
A Nearoptimal Algorithm for Learning Margin Halfspaces with Massart Noise
I. Diakonikolas, N. Zarifis
Advances in Neural Information Processing Systems (NeurIPS 2024)
Selected for Spotlight Presentation

Reliable Learning of Halfspaces under Gaussian Marginals
I. Diakonikolas, L. Ren, N. Zarifis
Advances in Neural Information Processing Systems (NeurIPS 2024)
Selected for Spotlight Presentation

Sample and Computationally Efficient Robust Learning of Gaussian SingleIndex Models
P. Wang, N. Zarifis, I. Diakonikolas, J. Diakonikolas
Advances in Neural Information Processing Systems (NeurIPS 2024)

Active Learning of General Halfspaces: Label Queries vs Membership Queries
I. Diakonikolas, D. Kane, M. Ma
Advances in Neural Information Processing Systems (NeurIPS 2024)

Learning a Single Neuron Robustly to Distributional Shifts and Adversarial Label Noise
S. Li, S. Karmalkar, I. Diakonikolas, J. Diakonikolas
Advances in Neural Information Processing Systems (NeurIPS 2024)

SumofSquares Lower Bounds for NonGaussian Component Analysis
I. Diakonikolas, S. Karmalkar, S. Pang, A. Potechin
Proceedings of the 65th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2024)

Agnostically Learning Multiindex Models with Queries
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 65th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2024)
Invited to SIAM Journal on Computing Special Issue for FOCS 2024
We study the power of query access for the fundamental task of
agnostic learning under the Gaussian distribution.
In the agnostic model, no assumptions are
made on the labels of the examples and the goal is to compute a hypothesis
that is competitive with the {\em bestfit} function in a known class, i.e.,
it achieves error $\opt+\eps$, where $\opt$
is the error of the best function in the class.
We focus on a general family of MultiIndex Models (MIMs),
which are $d$variate functions that depend only on few relevant directions,
i.e., have the form $g(\vec W \x)$ for an
unknown link function $g$ and a $k \times d$ matrix $\vec W$.
Multiindex models cover a wide range of commonly studied function classes,
including realvalued function classes
such as constantdepth neural networks with ReLU activations,
and Boolean concept classes such as intersections of halfspaces.
Our main result shows that query access gives significant runtime improvements
over random examples for agnostically learning
both realvalued and Booleanvalued MIMs.
Under standard regularity assumptions for the link function
(namely, bounded variation or surface area),
we give an agnostic query learner for MIMs
with running time $O(k)^{\poly(1/\eps)} \; \poly(d) $.
In contrast, algorithms that rely only on random labeled examples
inherently require $d^{\poly(1/\epsilon)}$ samples and runtime,
even for the basic problem of agnostically
learning a single ReLU or a halfspace.
As special cases of our general approach,
we obtain the following results:
For the class of depth$\ell$, width$S$ ReLU networks on
$\R^d$, our agnostic query learner runs in time $\poly(d) 2^{\poly(\ell S/\eps)}$.
This bound qualitatively matches the runtime of an algorithm by~\cite{CKM22}
for the realizable PAC setting with random examples.
For the class of arbitrary intersections of $k$ halfspaces on $\R^d$,
our agnostic query learner runs in time $\poly(d) \, 2^{\poly(\log (k)/\eps)}$.
Prior to our work, no improvement over the agnostic PAC model complexity
(without queries) was known, even for the case of a single halfspace.
\end{itemize}
In both these settings, we provide evidence
that the $2^{\poly(1/\eps)}$ runtime dependence is required
for proper query learners, even for agnostically learning a single ReLU or halfspace.
In summary, our algorithmic result establishes a strong computational
separation between
the agnostic PAC and the agnostic PAC+Query models under the Gaussian distribution.
Prior to our work, no such separation was known for any natural concept classes —
even for the special case of agnostically learning a single halfspace,
for which it was an open problem first posed by Feldman~\cite{Feldman08}.
Our results are enabled by a general dimensionreduction technique
that leverages query access to estimate gradients of (a smoothed version of)
the underlying label function.

Clustering Mixtures of Bounded Covariance Distributions Under Optimal Separation
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, J. Lee, T. Pittas
Proceedings of the 36th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2025)
We study the clustering problem for mixtures of bounded covariance distributions,
under a finegrained separation assumption. Specifically, given samples from a
$k$component mixture distribution $D = \sum_{i =1}^k w_i P_i$,
where each $w_i \ge \alpha$ for some known parameter $\alpha$,
and each $P_i$ has unknown covariance $\Sigma_i \preceq \sigma^2_i \cdot I_d$
for some unknown $\sigma_i$, the goal is to cluster the samples
assuming a pairwise mean separation in the order of $(\sigma_i+\sigma_j)/\sqrt{\alpha}$
between every pair of components $P_i$ and $P_j$. Our main contributions are as follows:
For the special case of nearly uniform mixtures, we give the first
polynomialtime algorithm for this clustering task.
Prior work either required separation scaling with the maximum cluster standard deviation
(i.e.~$\max_i \sigma_i$)~\cite{diakonikolas2022clustering} or required both additional
structural assumptions and mean separation scaling as a large degree polynomial in $1/\alpha$~\cite{BKK22}.
For arbitrary (i.e.~generalweight) mixtures, we point out that accurate
clustering is informationtheoretically impossible under our finegrained mean
separation assumptions. We introduce the notion of a {\em clustering refinement}
 a list of nottoosmall subsets satisfying a similar separation,
and which can be merged into a clustering approximating the
ground truth  and show that it is possible to efficiently compute an accurate
clustering refinement of the samples.
Furthermore, under a variant of the ``no large
subcluster'' condition introduced in prior work~\cite{BKK22},
we show that our algorithm will output an accurate clustering, not just a refinement, even for
generalweight mixtures.
As a corollary, we obtain efficient clustering algorithms for mixtures
of wellconditioned highdimensional logconcave distributions.
Moreover, our algorithm is robust to a fraction of adversarial outliers
comparable to $\alpha$.
At the technical level, our algorithm proceeds by first
using listdecodable mean estimation
to generate a polynomialsize list of possible cluster means,
before successively pruning candidates using
a carefully constructed convex program.
In particular, the convex program takes as input
a candidate mean $\muhat$ and a scale parameter $\hat{s}$,
and determines the existence
of a subset of points that could plausibly form a cluster
with scale $\hat{s}$ centered around $\muhat$.
While the natural way of designing this program makes it nonconvex,
we construct a convex relaxation which remains satisfiable
by (and only by) nottoosmall subsets of true clusters.

Fast CoTraining under Weak Dependence via StreamBased Active Learning
[abstract]
I. Diakonikolas, M. Ma, L. Ren, C. Tzamos
Proceedings of the 41st International Conference on Machine Learning (ICML 2024)
Selected for Oral Presentation
Cotraining is a classical semisupervised learning method
which only requires a small number of labeled examples
for learning, under reasonable assumptions.
Despite extensive literature on the topic,
very few hypothesis classes are known
to be provably efficiently learnable via cotraining,
even under very strong distributional assumptions.
In this work, we study the cotraining problem
in the streambased active learning model. We show that
a range of natural concept classes are efficiently learnable
via cotraining, in terms of both label efficiency
and computational efficiency.
We provide an efficient reduction of cotraining
under the standard assumption of weak dependence, in the streambased active model,
to online classification.
As a corollary, we obtain efficient cotraining algorithms
with {\em error independent} label complexity
for every concept class class efficiently learnable
in the mistake bound online model.
Our framework also gives cotraining algorithms
with label complexity $\Tilde{O}(d\log (1/\eps))$ for any concept
class with VC dimension $d$, though in general this reduction is not computationally efficient.
Finally, using additional ideas from online learning,
we design the first efficient cotraining algorithms with label complexity
$\Tilde{O}(d^2\log (1/\eps))$ for several concept classes, including
unions of intervals and homogeneous halfspaces.

Robust Sparse Estimation for Gaussians with Optimal Error under Huber Contamination
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, S. Karmalkar, A. Pensia, T. Pittas
Proceedings of the 41st International Conference on Machine Learning (ICML 2024)
We study Gaussian sparse estimation tasks in Huber's contamination model with a focus
on mean estimation, PCA, and linear regression. For each of these tasks, we give
the first sample and computationally efficient robust estimators with optimal error
guarantees, within constant factors. All prior efficient algorithms for these
tasks incur quantitatively suboptimal error. Concretely, for Gaussian robust
$k$sparse mean estimation on $\R^d$ with corruption rate $\eps>0$, our
algorithm
has sample complexity $(k^2/\eps^2)\polylog(d/\eps)$, runs in sample polynomial time,
and approximates the target mean within $\ell_2$error $O(\eps)$. Previous efficient
algorithms inherently incur error $\Omega(\eps \sqrt{\log(1/\eps)})$.
At the technical level, we develop a novel multidimensional filtering method
in the sparse regime that may find other applications.

Robustly Learning SingleIndex Models via Alignment Sharpness
[abstract]
[arxiv]
N. Zarifis, P. Wang, I. Diakonikolas, J. Diakonikolas
Proceedings of the 41st International Conference on Machine Learning (ICML 2024)
We study the problem of learning SingleIndex Models
under the $L_2^2$ loss in the agnostic model.
We give an efficient learning algorithm,
achieving a constant factor approximation to the optimal loss,
that succeeds under a range of distributions
(including logconcave distributions) and a broad class of
monotone and Lipschitz link functions.
This is the first efficient constant factor approximate agnostic learner, even for Gaussian data
and for any nontrivial class of link functions. Prior
work for the case of unknown link function either works in
the realizable setting or does not attain constant factor approximation. The main technical
ingredient enabling our algorithm and analysis
is a novel notion of a
local error bound in optimization
that we term {\em alignment sharpness}
and that may be of broader interest.

Testable Learning of General Halfspaces with Adversarial Label Noise
I. Diakonikolas, D. Kane, S. Liu, N. Zarifis
Proceedings of the 37th Annual Conference on Learning Theory (COLT 2024)

Statistical Query Lower Bounds for Learning Truncated Gaussians
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, T. Pittas, N. Zarifis
Proceedings of the 37th Annual Conference on Learning Theory (COLT 2024)
We study the problem of estimating the mean of an identity covariance Gaussian in the
truncated setting, in the regime when the truncation set comes from a lowcomplexity
family $\cal{C}$ of sets. Specifically, for a fixed but unknown truncation set
$S \subseteq \R^d$, we are given access to samples from the distribution
$\cN(\bm \mu, \vec I)$ truncated to the set $S$. The goal is to estimate $\bm \mu$
within accuracy $\eps>0$ in $\ell_2$norm. Our main result is a Statistical Query (SQ)
lower bound suggesting a superpolynomial informationcomputation gap for this task. In
more detail, we show that the complexity of any SQ algorithm for this problem is
$d^{\poly(1/\eps)}$, even when the class $\cal{C}$ is simple so that $\poly(d/\eps)$
samples informationtheoretically suffice. Concretely, our SQ lower bound applies when
$\cal{C}$ is a union of a bounded number of rectangles whose VC dimension and Gaussian
surface are small. As a corollary of our construction, it also
follows that the complexity of the previously known algorithm for this task
is qualitatively best possible.

Efficiently Learning OneHiddenLayer ReLU Networks via Schur Polynomials
[abstract]
[arxiv]
I. Diakonikolas, D. Kane *Author order randomized
Proceedings of the 37th Annual Conference on Learning Theory (COLT 2024)
We study the problem of PAC learning a linear combination of $k$ ReLU activations
under the standard Gaussian distribution on $\R^d$ with respect to the square loss.
Our main result is an efficient algorithm for this learning task with sample and
computational complexity $(dk/\eps)^{O(k)}$, where $\eps>0$ is the target accuracy.
Prior work had given an algorithm for this problem with complexity $(dk/\eps)^{h(k)}$,
where the function $h(k)$ scales superpolynomially in $k$. Interestingly,
the complexity of our algorithm is nearoptimal within the class of
Correlational Statistical Query algorithms.
At a highlevel, our algorithm uses tensor decomposition to identify a subspace
such that all the $O(k)$order moments are small in the orthogonal directions.
Its analysis makes essential use of the theory of Schur polynomials
to show that the highermoment error tensors are small given that the lowerorder ones are.

Super Nonsingular Decompositions of Polynomials and their Application to
Robustly Learning Lowdegree PTFs
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Kontonis, S. Liu, N. Zarifis
Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024)
We study the efficient learnability of lowdegree polynomial threshold functions
(PTFs) in the presence of a constant fraction of adversarial corruptions.
Our main algorithmic result is a polynomialtime PAC learning algorithm for this
concept class in the strong contamination model under the Gaussian distribution
with error guarantee $O_{d, c}(\opt^{1c})$, for any desired constant $c>0$,
where $\opt$ is the fraction of corruptions.
In the strong contamination model, an omniscient adversary
can arbitrarily corrupt an $\opt$fraction
of the data points and their labels.
This model generalizes the malicious noise model and the adversarial label noise
model. Prior to our work, known polynomialtime algorithms
in this corruption model (or even in the weaker adversarial label noise model)
achieved error $\tilde{O}_d(\opt^{1/(d+1)})$, which
deteriorates significantly as a function of the degree $d$.
Our algorithm employs an iterative approach inspired by localization techniques
previously used in the context of learning linear threshold functions.
Specifically, we use a robust perceptron algorithm to compute
a good partial classifier and then iterate on the unclassified points.
In order to achieve this, we need to take a set defined by a number
of polynomial inequalities and partition it into several wellbehaved subsets.
To this end, we develop new polynomial decomposition techniques
that may be of independent interest.

Testing Closeness of Multivariate Distributions via Ramsey Theory
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, S. Liu
Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024)
We investigate the statistical task of closeness (or equivalence) testing for multidimensional
distributions. Specifically, given sample access to two unknown distributions $\mathbf p, \mathbf q$
on $\R^d$, we want to distinguish between the case that
$\mathbf p=\mathbf q$ versus $\\mathbf p\mathbf q\_{\mathcal A_k} > \epsilon$,
where $\\mathbf p\mathbf q\_{\mathcal A_k}$ denotes the generalized $\mathcal A_k$ distance
between $\mathbf p$ and $\mathbf q$ 
measuring the maximum discrepancy between the distributions over any collection of $k$ disjoint,
axisaligned rectangles. Our main result is the first closeness tester for this problem
with {\em sublearning} sample complexity in any fixed dimension and a
nearlymatching sample complexity lower bound.
In more detail, we provide a computationally efficient closeness tester with
sample complexity $O\left((k^{6/7}/ \mathrm{poly}_d(\epsilon)) \log^d(k)\right)$.
On the lower bound side, we establish a qualitatively matching sample complexity lower bound
of $\Omega(k^{6/7}/\mathrm{poly}(\epsilon))$, even for $d=2$. These sample complexity bounds
are surprising because the sample complexity of the problem in the univariate setting
is $\Theta(k^{4/5}/\mathrm{poly}(\epsilon))$. This has the interesting consequence that
the jump from one to two dimensions leads to a substantial increase in sample complexity,
while increases beyond that do not.
As a corollary of our general $\mathcal A_k$ tester,
we obtain $d_{\mathrm TV}$closeness testers for
pairs of $k$histograms on $\R^d$
over a common unknown partition, and pairs of uniform distributions
supported on the union of $k$ unknown disjoint axisaligned rectangles.
Both our algorithm and our lower bound
make essential use of tools from Ramsey theory.

How Does Wild Data Provably Help OOD Detection?
[abstract]
[arxiv]
X. Du, Z. Fang, I. Diakonikolas, Y. Li
Proceedings of the 12th International Conference on Learning Representations (ICLR 2024)
Using unlabeled data to regularize the machine learning models has demonstrated promise
for improving safety and reliability in detecting outofdistribution (OOD) data.
Harnessing the power of unlabeled inthewild data is nontrivial due to the heterogeneity
of both indistribution (ID) and OOD data. This lack of a clean set of OOD samples poses significant challenges
in learning an optimal OOD classifier. Currently, there is a lack of research on formally understanding
how unlabeled data helps OOD detection. This paper bridges the gap by introducing a new learning
framework that offers both strong theoretical guarantees and empirical effectiveness.
The framework separates candidate outliers from the unlabeled data and then trains an
OOD classifier using the candidate outliers and the labeled ID data.
Theoretically, we provide rigorous error bounds from the lens of separability and
learnability, formally justifying the two components in our algorithm. Our theory shows that our framework
can separate the candidate outliers with small error rates,
which leads to a generalization guarantee for the learned OOD classifier.
Empirically, our method achieves stateoftheart performance on common benchmarks,
reinforcing our theoretical insights.

Online Robust Mean Estimation
[abstract]
[arxiv]
D. Kane, I. Diakonikolas, H. Xiao, S. Liu *Author order randomized
Proceedings of the 35th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2024)
We study the problem of highdimensional robust mean estimation in an online setting.
Specifically, we consider a scenario where $n$ sensors are measuring some common, ongoing phenomenon.
At each time step $t=1,2,\ldots,T$, the $i^{th}$ sensor reports its readings $x^{(i)}_t$ for that time step.
The algorithm must then commit to its estimate $\mu_t$ for the true mean value
of the process at time $t$. We assume that most of the sensors observe
independent samples from some common distribution $X$,
but an $\eps$fraction of them may instead behave maliciously.
The algorithm wishes to compute a good approximation $\mu$ to the true mean $\mu^\ast := \E[X]$.
We note that if the algorithm is allowed to wait until time $T$ to report its estimate,
this reduces to the wellstudied problem of robust mean estimation.
However, the requirement that our algorithm produces partial estimates
as the data is coming in substantially complicates the situation.
We prove two main results about online robust mean estimation in this model.
First, if the uncorrupted samples satisfy the standard condition of $(\eps,\delta)$stability,
we give an efficient online algorithm that outputs estimates $\mu_t$, $t \in [T],$
such that with high probability it holds that $\\mu\mu^\ast\_2 = O(\delta \log(T))$,
where $\mu = (\mu_t)_{t \in [T]}$.
We note that this error bound is nearly competitive with the best offline algorithms,
which would achieve $\ell_2$error of $O(\delta)$.
Our second main result shows that
with additional assumptions on the input (most notably that $X$ is a product distribution)
there are inefficient algorithms whose error does not depend on $T$ at all.

First Order Stochastic Optimization with Oblivious Noise
[abstract]
I. Diakonikolas, S. Karmalkar, J. Park, C. Tzamos
Advances in Neural Information Processing Systems (NeurIPS 2023)
We initiate the study of stochastic optimization with oblivious noise,
broadly generalizing the standard heavytailed noise setup.
In our setting, in addition to random observation noise,
the stochastic gradient
may be subject to independent \emph{oblivious noise},
which may not have bounded moments and is not necessarily centered.
Specifically, we assume access to a noisy oracle for the stochastic gradient of $f$
at $x$, which returns a vector $\nabla f(\gamma, x) + \xi$, where $\gamma$ is
the bounded variance observation noise
and $\xi$ is the oblivious noise that is independent of $\gamma$ and $x$.
The only assumption we make on the oblivious noise $\xi$
is that $\Pr[\xi = 0] \ge \alpha$ for some $\alpha \in (0, 1)$.
In this setting, it is not informationtheoretically possible to recover a single solution
close to the target when the fraction of inliers $\alpha$ is less than $1/2$.
Our main result is an efficient {\em listdecodable} learner that recovers
a small list of candidates at least one of which is close to the true solution.
On the other hand, if $\alpha = 1\eps$, where $0< \eps< 1/2$ is sufficiently small
constant, the algorithm recovers a single solution.
Along the way, we develop a rejectionsamplingbased algorithm to
perform noisy location estimation, which may be of independent interest.

SQ Lower Bounds for NonGaussian Component Analysis with Weaker Assumptions
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, L. Ren, Y. Sun
Advances in Neural Information Processing Systems (NeurIPS 2023)
We study the complexity of NonGaussian Component Analysis (NGCA) in
the Statistical Query (SQ) model. Prior work developed a methodology
to prove SQ lower bounds for NGCA that have been applicable to a wide range
of contexts. In particular, it was known that for any univariate
distribution $A$ satisfying certain conditions, distinguishing between
a standard multivariate Gaussian and a distribution that behaves like
$A$ in a random hidden direction and like a standard Gaussian in the orthogonal
complement, is SQhard. The required conditions were that (1) $A$
matches many loworder moments with a standard Gaussian, and
(2) the chisquared norm of $A$ with respect to the standard Gaussian
is finite. While the momentmatching condition is necessary for
hardness, the chisquared condition was only required for technical reasons.
In this work, we establish that the latter condition is indeed not necessary.
In particular, we prove nearoptimal SQ lower bounds for NGCA
under the momentmatching condition only. Our SQ lower bound result can be
generalized to the setting where the hidden subspace is multidimensional.

SQ Lower Bounds for Learning Mixtures of Linear Classifiers
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, Y. Sun
Advances in Neural Information Processing Systems (NeurIPS 2023)
We study the problem of learning mixtures of linear
classifiers under Gaussian covariates.
Given sample access to a mixture of $r$ distributions on $\R^n$
of the form $(\bx,y_{\ell})$, $\ell \in [r]$, where
$\bx\sim\mathcal{N}(0,\mathbf{I}_n)$ and $y_\ell=\sgn(\langle\bv_\ell,\bx\rangle)$
for an unknown unit vector $\bv_\ell$, the goal is to learn the underlying
distribution in total variation distance. Our main result is a
Statistical Query (SQ) lower bound suggesting that known
algorithms for this problem are essentially best
possible, even for the special case of uniform mixtures.
In particular, we show that the complexity of any SQ algorithm
for the problem is $n^{\poly(1/\Delta) \log(r)}$,
where $\Delta$ is a lower bound on the pairwise $\ell_2$separation
between the $\bv_\ell$'s. The key technical
ingredient underlying our result is a new construction of
spherical designs that may be of independent interest.

NearOptimal Algorithms for Gaussians with Huber Contamination:
Mean Estimation and Linear Regression
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Pensia, T. Pittas
Advances in Neural Information Processing Systems (NeurIPS 2023)
We study the fundamental problems of Gaussian mean
estimation and linear regression with Gaussian covariates
in the presence of Huber contamination. Our main
contribution is the design of the first sample nearoptimal
and almost lineartime algorithms with optimal error
guarantees for both these problems. Specifically, for
Gaussian robust mean estimation on $\R^d$ with
contamination parameter $\eps \in (0, \eps_0)$ for a small
absolute constant $\eps_0$, we give an
algorithm with sample complexity $n = \tilde{O}(d/\eps^2)$
and almost linear runtime that approximates the target
mean within $\ell_2$error $O(\eps)$.
This improves on
prior work that achieved this error guarantee with
polynomially suboptimal sample and time complexity.
For robust linear
regression, we give the first algorithm with sample
complexity $n = \tilde{O}(d/\eps^2)$ and almost linear
runtime that approximates the target regressor within
$\ell_2$error $O(\eps)$. This is the first polynomial
sample and time algorithm achieving the optimal error
guarantee, answering an open question in the literature.
At the technical level, we develop a methodology that
yields almostlinear time algorithms for multidirectional
filtering that may be of broader interest.

Robust SecondOrder Nonconvex Optimization and Its Application to LowRank Matrix Sensing
[abstract]
S. Li, Y. Cheng, I. Diakonikolas, J. Diakonikolas, R. Ge, S. Wright
Advances in Neural Information Processing Systems (NeurIPS 2023)
Finding an approximate secondorder stationary point (SOSP)
is a fundamental problem in stochastic nonconvex optimization
with many applications in machine learning.
However, this problem is poorly understood in the presence of outliers,
limiting the use of existing nonconvex algorithms in adversarial settings.
In this paper, we study the problem of finding SOSPs in the strong contamination model,
where a constant fraction of the datapoints are arbitrarily corrupted.
We develop an efficient algorithm that (under mild assumptions)
outputs an approximate SOSP with \emph{dimensionindependent} accuracy guarantees,
using $\widetilde{O}(\frac{D^2}{\epsilon})$ samples
where $D$ is the ambient dimension and $\epsilon$ is the fraction of corrupted datapoints.
As our main application, we show that lowrank matrix sensing
can be robustly solved in our framework, allowing for corruptions in both the sensing matrices
and the measurements.
On the lower bound side, we establish a Statistical Query lower bound
providing evidence that the quadratic dependence on $D$
in the sample complexity is necessary for computationally efficient algorithms.

NearOptimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise
[abstract]
[arxiv]
I. Diakonikolas, J. Diakonikolas, D. Kane, P. Wang, N. Zarifis
Advances in Neural Information Processing Systems (NeurIPS 2023)
We study the problem of learning general (i.e., not necessarily homogeneous)
halfspaces with Random Classification Noise under the Gaussian distribution.
We establish nearlymatching algorithmic and Statistical Query (SQ) lower bound results
revealing a surprising informationcomputation gap for this basic problem.
Specifically, the sample complexity of this learning problem is
$\widetilde{\Theta}(d/\eps)$, where $d$ is the dimension and $\eps$ is the excess error.
Our positive result is a computationally efficient learning algorithm with sample complexity
$\tilde{O}(d/\eps + d/(\max\{p, \eps\})^2)$,
where $p$ quantifies the bias of the target halfspace.
On the lower bound side, we show that any efficient SQ algorithm (or lowdegree test)
for the problem requires sample complexity at least
$\Omega(d^{1/2}/(\max\{p, \eps\})^2)$.
Our lower bound suggests that this quadratic dependence on $1/\eps$
is inherent for efficient algorithms.

Efficient Testable Learning of Halfspaces with Adversarial Label Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Kontonis, S. Liu, N. Zarifis
Advances in Neural Information Processing Systems (NeurIPS 2023)
We give the first polynomialtime algorithm for the testable learning
of halfspaces in the presence of adversarial label noise under the
Gaussian distribution. In the recently introduced testable learning
model, one is required to produce a
testerlearner such that if the data passes the tester,
then one can trust the output of the robust learner on the data.
Our testerlearner runs in time $\poly(d/\eps)$ and outputs a
halfspace with misclassification error $O(\opt)+\eps$, where $\opt$ is
the 01 error of the best fitting halfspace.
At a technical level, our algorithm employs an iterative soft localization technique
enhanced with appropriate testers to ensure that the data distribution is sufficiently
similar to a Gaussian.

A Spectral Algorithm for ListDecodable Covariance Estimation in Relative Frobenius Norm
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, J. Lee, A. Pensia, T. Pittas
Advances in Neural Information Processing Systems (NeurIPS 2023)
Selected for Spotlight Presentation
We study the problem of listdecodable Gaussian covariance estimation.
Given a multiset $T$ of $n$ points in $\R^d$ such that an unknown
$\alpha<1/2$ fraction of points in $T$ are i.i.d. samples from an unknown
Gaussian $\mathcal{N}(\mu, \Sigma)$, the goal is to output
a list of $O(1/\alpha)$ hypotheses at least one of which is close to $\Sigma$
in relative Frobenius norm.
Our main result is a $\poly(d,1/\alpha)$ sample and time algorithm
for this task that guarantees relative Frobenius norm error of $\poly(1/\alpha)$.
Importantly, our algorithm relies purely on spectral techniques.
As a corollary, we obtain an efficient spectral algorithm
for robust partial clustering of Gaussian mixture
models (GMMs)  a key ingredient in the recent work of~\cite{BakDJKKV22}
on robustly learning arbitrary GMMs.
Combined with the other components of~\cite{BakDJKKV22}, our new method
yields the first SumofSquaresfree algorithm
for robustly learning GMMs, resolving an open problem proposed by Vempala and Kothari.
At the technical level, we develop a novel multifiltering method
for listdecodable covariance estimation that may be useful in other settings.

DistributionIndependent Regression for Generalized Linear Models with Oblivious Corruptions
[abstract]
[arxiv]
I. Diakonikolas, S. Karmalkar, J. Park, C. Tzamos
Proceedings of the 36th Annual Conference on Learning Theory (COLT 2023)
We demonstrate the first algorithms for the problem of regression for generalized linear models (GLMs)
in the presence of additive oblivious noise.
We assume we have sample access to examples $(x, y)$ where $y$ is a noisy measurement of $g(w^* \cdot x)$.
In particular, the noisy labels are of the form $y = g(w^* \cdot x) + \xi + \eps$,
where $\xi$ is the oblivious noise drawn independently of $x$ and satisfies
$\Pr[\xi = 0] \geq o(1)$, and $\eps \sim \cN(0, \sigma^2)$.
Our goal is to accurately recover a parameter vector $w$ such that the function $g(w \cdot x)$
has arbitrarily small error when compared to the true values $g(w^* \cdot x)$, rather than the noisy measurements $y$.
We present an algorithm that tackles this problem in its most general distributionindependent setting,
where the solution may not even be identifiable. Our algorithm returns an accurate estimate of the solution
if it is identifiable, and otherwise returns a small list of candidates, one of which is close to the true solution.
Furthermore, we provide a necessary and sufficient condition for identifiability, which holds in broad settings.
Specifically, the problem is identifiable when the quantile at which $\xi + \eps = 0$ is known,
or when the family of hypotheses does not contain candidates that are nearly equal to a translated $g(w^* \cdot x) + A$
for some real number $A$, while also having large error when compared to $g(w^* \cdot x)$.
This is the first algorithmic result for GLM regression with oblivious noise which can handle more than half the samples
being arbitrarily corrupted. Prior work focused largely on the setting of linear regression, and gave algorithms
under restrictive assumptions.

SelfDirected Linear Classification
[abstract]
[arxiv]
I. Diakonikolas, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 36th Annual Conference on Learning Theory (COLT 2023)
In online classification, a learner is presented with a sequence of examples
and aims to predict their labels in an online fashion so as to minimize
the total number of mistakes.
In the selfdirected variant, the learner knows in advance
the pool of examples and can adaptively choose the order in which predictions are made.
Here we study the power of choosing the prediction order and establish
the first strong separation between worstorder and randomorder learning
for the fundamental task of linear classification.
Prior to our work, such a separation was known only for very restricted concept classes,
e.g., onedimensional thresholds or axisaligned rectangles.
We present two main results.
If $X$ is a dataset of $n$ points drawn uniformly at random from the $d$dimensional unit sphere,
we design an efficient selfdirected learner that
makes $O(d \log \log(n))$ mistakes and classifies the entire dataset.
If $X$ is an arbitrary $d$dimensional dataset of size $n$,
we design an efficient selfdirected learner that predicts the labels
of $99\%$ of the points in $X$ with mistake bound independent of $n$.
In contrast, under a worst or randomordering, the number of mistakes
must be at least $\Omega(d \log n)$, even when the
points are drawn uniformly from the unit sphere
and the learner only needs to predict the labels for $1\%$ of them.

InformationComputation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise
[abstract]
[arxiv]
I. Diakonikolas, J. Diakonikolas, D. Kane, P. Wang, N. Zarifis
Proceedings of the 36th Annual Conference on Learning Theory (COLT 2023)
We study the problem of PAC learning $\gamma$margin halfspaces
with Random Classification Noise.
We establish an informationcomputation tradeoff
suggesting an inherent gap between the sample complexity of the problem
and the sample complexity of computationally efficient algorithms.
Concretely, the sample complexity of the problem is $\widetilde{\Theta}(1/(\gamma^2 \eps))$.
We start by giving a simple efficient algorithm
with sample complexity $\widetilde{O}(1/(\gamma^2 \eps^2))$.
Our main result is a lower bound for Statistical Query (SQ) algorithms
and lowdegree polynomial tests suggesting that the quadratic dependence on $1/\eps$
in the sample complexity is inherent for computationally efficient algorithms.
Specifically, our results imply a lower bound of $\widetilde{\Omega}(1/(\gamma^{1/2} \eps^2))$
on the sample complexity of any efficient SQ learner or lowdegree test.

SQ Lower Bounds for Learning Mixtures
of Separated and Bounded Covariance Gaussians
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, T. Pittas, N. Zarifis
Proceedings of the 36th Annual Conference on Learning Theory (COLT 2023)
We study the complexity of learning mixtures of separated Gaussians with common unknown bounded
covariance matrix. Specifically, we focus on learning Gaussian mixture models (GMMs) on $\R^d$
of the form $P= \sum_{i=1}^k w_i N(\mu_i,\Sigma_i)$,
where $\Sigma_i = \Sigma \preceq I$
and $\min_{i \neq j} \\mu_i \mu_j\_2 \geq k^\eps$ for some $\eps>0$.
Known learning algorithms for this family of GMMs have complexity $(dk)^{O(1/\eps)}$. In this work, we
prove that any Statistical Query (SQ) algorithm for this problem requires complexity at least
$d^{\Omega(1/\eps)}$. In the special case where the separation is on the order of $k^{1/2}$,
we additionally obtain finegrained SQ lower bounds with the correct exponent.
Our SQ lower bounds imply similar lower bounds for lowdegree polynomial tests.
Conceptually, our results provide evidence that known algorithms for
this problem are nearly best possible.

Statistical and Computational Limits for TensoronTensor
Association Detection
[abstract]
[conference]
I. Diakonikolas, D. Kane, Y. Luo, A. Zhang
Proceedings of the 36th Annual Conference on Learning Theory (COLT 2023)
In this paper, we consider the tensorontensor association detection problem,
where the goal is to detect whether there is an association between
the tensor responses to tensor covariates linked via a lowrank tensor parameter.
We first develop tight bounds on the signaltonoise ratio (SNR)
such that the detection problem is statistically possible.
We then provide testing procedures that succeed when the SNR is above the threshold.
On the other hand, the statistical optimal tests often require computing the
largest singular value of a given tensor, which can be NPhard in general.
To complement that, we develop efficient polynomialtime testing procedures
with provable guarantees. We also develop matching lower bounds under
the Statistical Query model and show that the SNRs required by the proposed
polynomialtime algorithms are essential for computational efficiency.
We identify a gap that appears between the SNR requirements
of the optimal unconstrainedtime tests and polynomialtime tests
if and only if the sum of the tensor response order and
the tensor covariate order is no less than three. To our best knowledge,
this is the first complete characterization of the statistical
and computational limits for the general tensorontensor association
detection problem. Our findings significantly generalize the
results in the literature on signal detection in linear regression
and lowrank matrix trace regression.

A Nearly Tight Bound for Fitting an Ellipsoid to Gaussian Random Points
[abstract]
[arxiv]
D. Kane, I. Diakonikolas *Author order randomized
Proceedings of the 36th Annual Conference on Learning Theory (COLT 2023)
We prove that for $c>0$ a sufficiently small universal constant
that a random set of $cd^2/log^4(d)$ independent Gaussian random points
in $\R^d$ lie on a common ellipsoid with high probability.
This nearly establishes a conjecture of~\cite{SaundersonCPW12}, within logarithmic factors.
The latter conjecture has attracted significant attention over the past decade,
due to its connections to machine learning and sumofsquares lower bounds
for certain statistical problems.

Robustly Learning a Single Neuron via Sharpness
[abstract]
[arxiv]
P. Wang, N. Zarifis, I. Diakonikolas, J. Diakonikolas
Proceedings of the 40th International Conference on Machine Learning (ICML 2023)
Selected for Oral Presentation
We study the problem of learning a single neuron
with respect to the $L_2^2$loss in the presence
of adversarial label noise.
We give an efficient algorithm that,
for a broad family of activations including ReLUs,
approximates the optimal $L_2^2$error within a constant factor.
Our algorithm applies under much milder distributional assumptions
compared to prior work.
The key ingredient enabling our results
is a novel connection to local error bounds
from optimization theory.

NearOptimal Cryptographic Hardness of Agnostically Learning Halfspaces
and ReLU Regression under Gaussian Marginals
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, L. Ren
Proceedings of the 40th International Conference on Machine Learning (ICML 2023)
We study the task of agnostically learning halfspaces under the Gaussian distribution.
Specifically, given labeled examples $(\bx,y)$ from an unknown distribution
on $\R^n \times \{ \pm 1\}$,
whose marginal distribution on $\bx$ is the standard Gaussian
and the labels $y$ can be arbitrary,
the goal is to output a hypothesis with 01 loss $\opt+\eps$,
where $\opt$ is the 01 loss of the bestfitting halfspace.
We prove a nearoptimal computational hardness result for this task,
under the widely believed
subexponential time hardness of the Learning with Errors (LWE) problem.
Prior hardness results are either
qualitatively suboptimal or apply to restricted families of algorithms.
Our techniques extend to
yield nearoptimal lower bounds for related problems,
including ReLU regression.

NearlyLinear Time and Streaming Algorithms for OutlierRobust PCA
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Pensia, T. Pittas
Proceedings of the 40th International Conference on Machine Learning (ICML 2023)
We study principal component analysis (PCA), where given a dataset in $\R^d$ from a distribution,
the task is to find a unit vector $v$ that approximately maximizes the variance
of the distribution after being projected along $v$.
Despite being a classical task, standard estimators fail drastically if the data contains
even a small fraction of outliers,
motivating the problem of robust PCA.
Recent work has developed computationallyefficient algorithms for robust PCA
that either take superlinear time or have suboptimal error guarantees.
Our main contribution is to develop a nearlylinear time algorithm for robust PCA
with nearoptimal error guarantees. We also develop a singlepass streaming algorithm
for robust PCA with memory usage nearlylinear in the dimension.

A Strongly Polynomial Algorithm for Approximate Forster Transforms
and its Application to Halfspace Learning
[abstract]
[arxiv]
I. Diakonikolas, C. Tzamos, D. Kane *Author order randomized
Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC 2023)
The Forster transform is a method of regularizing a dataset
by placing it in {\em radial isotropic position} while maintaining
some of its essential properties. Forster transforms have played
a key role in a diverse range of settings spanning computer science
and functional analysis. Prior work had given {\em weakly} polynomial time algorithms
for computing Forster transforms, when they exist. Our main result is the
first {\em strongly polynomial time} algorithm to compute an approximate
Forster transform of a given dataset or certify that no such transformation exists.
By leveraging our strongly polynomial Forster algorithm, we obtain the first
strongly polynomial time algorithm for {\em distributionfree} PAC learning of halfspaces.
This learning result is surprising because {\em proper} PAC learning of halfspaces
is {\em equivalent} to linear programming. Our learning approach extends
to give a strongly polynomial halfspace learner in the presence
of random classification noise and, more generally, Massart noise.

Gaussian Mean Testing Made Simple
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Pensia
SIAM Symposium on Simplicity in Algorithms (SOSA 2023)
We study the following fundamental hypothesis testing problem,
which we term {\em Gaussian mean testing}.
Given i.i.d. samples from a distribution $p$ on $\R^d$,
the task is to distinguish, with high probability,
between the following cases:
(i) $p$ is the standard Gaussian distribution, $N(0,I_d)$, and
(ii) $p$ is a Gaussian $N(\mu,\Sigma)$ for some unknown covariance $\Sigma$
and mean $\mu \in \R^d$ satisfying $\\mu\_2 \geq \epsilon$.
Recent work gave an algorithm for this testing problem with the
optimal sample complexity of $\Theta(\sqrt{d}/\epsilon^2)$.
Both the previous algorithm and its analysis are quite complicated.
Here we give an extremely simple algorithm for Gaussian mean testing
with a onepage analysis. Our algorithm is sample optimal
and runs in sample linear time.

Cryptographic Hardness of Learning Halfspaces with Massart Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, P. Manurangsi, L. Ren
Advances in Neural Information Processing Systems (NeurIPS 2022)
We study the complexity of PAC learning halfspaces in the presence of Massart noise.
In this problem, we are given i.i.d.\ labeled examples
$(\mathbf{x}, y) \in \R^N \times \{ \pm 1\}$,
where the distribution of $\mathbf{x}$ is arbitrary
and the label $y$ is a Massart corruption
of $f(\mathbf{x})$, for an unknown halfspace $f: \R^N \to \{ \pm 1\}$,
with flipping probability $\eta(\mathbf{x}) \leq \eta < 1/2$.
The goal of the learner is to compute a hypothesis with small 01 error.
Our main result is the first computational hardness result for this learning problem.
Specifically, assuming the (widely believed) subexponentialtime hardness
of the Learning with Errors (LWE) problem, we show that no polynomialtime
Massart halfspace learner can achieve error better than $\Omega(\eta)$,
even if the optimal 01 error is small, namely
$\opt = 2^{\log^{c} (N)}$
for any universal constant $c \in (0, 1)$.
Prior work had provided qualitatively similar evidence of hardness in the
Statistical Query model. Our computational hardness result
essentially resolves the polynomial PAC learnability of Massart halfspaces,
by showing that known efficient learning algorithms
for the problem are nearly best possible.

SQ Lower Bounds for Learning Single Neurons with Massart Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, L. Ren, Y. Sun
Advances in Neural Information Processing Systems (NeurIPS 2022)
We study the problem of PAC learning a single neuron in the presence of Massart noise.
Specifically, for a known activation function $f: \R \to \R$, the learner is given
access to labeled examples $(\bx, y) \in \R^d \times \R$,
where the marginal distribution of $\bx$ is arbitrary
and the corresponding label $y$ is a Massart corruption of $f(\langle \bw, \bx \rangle)$.
The goal of the learner is to output a hypothesis $h: \R^d \to \R$ with small squared loss.
For a range of activation functions, including ReLUs,
we establish superpolynomial Statistical Query (SQ) lower bounds for this learning problem.
In more detail, we prove that no efficient SQ algorithm can approximate
the optimal error within any constant factor.
Our main technical contribution is a novel SQhard construction
for learning $\{ \pm 1\}$weight Massart halfspaces on the Boolean hypercube
that is interesting on its own right.

NearlyTight Bounds for Testing Histogram Distributions
[abstract]
[arxiv]
C. Canonne, I. Diakonikolas, D. Kane, S. Liu
Advances in Neural Information Processing Systems (NeurIPS 2022)
We investigate the problem of testing
whether a discrete probability distribution over an ordered domain
is a histogram on a specified number of bins.
One of the most common tools for the succinct approximation of data,
{\em $k$histograms} over $[n]$, are probability distributions
that are piecewise constant over a set of $k$ intervals.
The histogram testing problem is the following:
Given samples from an unknown distribution $\p$ on $[n]$,
we want to distinguish between the cases that $\p$ is a $k$histogram
versus $\eps$far from any $k$histogram, in total variation distance.
Our main result is a sample nearoptimal and computationally efficient algorithm
for this testing problem, and a nearlymatching
(within logarithmic factors) sample complexity lower bound.
Specifically, we show that the histogram testing problem has sample complexity
$\widetilde \Theta (\sqrt{nk} / \eps + k / \eps^2 + \sqrt{n} / \eps^2)$.

OutlierRobust Sparse Mean Estimation for HeavyTailed Distributions
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, J. Lee, A. Pensia
Advances in Neural Information Processing Systems (NeurIPS 2022)
We study the fundamental task of outlierrobust mean estimation
for heavytailed distributions in the presence of sparsity.
Specifically, given a small number of corrupted samples
from a highdimensional heavytailed distribution
whose mean $\mu$ is guaranteed to be sparse,
the goal is to efficiently compute a hypothesis
that accurately approximates $\mu$ with high probability.
Prior work had obtained efficient algorithms
for robust sparse mean estimation of lighttailed distributions.
In this work, we give the first sampleefficient
and polynomialtime robust sparse mean estimator
for heavytailed distributions under mild moment assumptions.
Our algorithm achieves the optimal asymptotic error
using a number of samples scaling logarithmically with the ambient dimension.
Importantly, the sample complexity of our method is optimal
as a function of the failure probability $\tau$, having an {\em additive} $\log(1/\tau)$ dependence.
Our algorithm leverages the stabilitybased approach from the algorithmic robust statistics literature,
with crucial (and necessary) adaptations required in our setting.
Our analysis may be of independent interest,
involving the delicate design of a (nonspectral) decomposition
for positive semidefinite matrices satisfying certain sparsity properties.

ListDecodable Sparse Mean Estimation via DifferenceofPairs Filtering
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, S. Karmalkar, A. Pensia, T. Pittas
Advances in Neural Information Processing Systems (NeurIPS 2022)
Selected for Oral Presentation
We study the problem of listdecodable \emph{sparse} mean estimation.
Specifically, for a parameter $\alpha \in (0, 1/2)$,
we are given $m$ points in $\R^n$, $\lfloor \alpha m \rfloor$ of which
are i.i.d.\ samples from a distribution $D$ with unknown $k$sparse mean $\mu$.
No assumptions are made on the remaining points,
which form the majority of the dataset.
The goal is to return a small list of candidates
containing a vector $\hat \mu$ such that $\norm{\hat \mu  \mu}_2$ is small.
Prior work had studied the problem of listdecodable mean estimation in the dense setting.
In this work, we develop a novel, conceptually simpler technique
for listdecodable mean estimation. As the main application of our approach,
we provide the first sample and computationally efficient algorithm
for listdecodable sparse mean estimation. In particular, for distributions with
``certifiably bounded'' $t$th moments in $k$sparse directions
and sufficiently light tails, our algorithm achieves error of $(1/\alpha)^{O(1/t)}$
with sample complexity $m = (k\log(n))^{O(t)}/\alpha$ and running time $\poly(mn^t)$.
For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of
$\Theta (\sqrt{\log(1/\alpha)})$ with quasipolynomial
sample and computational complexity.
We complement our upper bounds with nearlymatching statistical query
and lowdegree polynomial testing lower bounds.

OutlierRobust Sparse Estimation via NonConvex Optimization
[abstract]
[arxiv]
Y. Cheng, I. Diakonikolas, R. Ge, S. Gupta, D. Kane, M. Soltanolkotabi
Advances in Neural Information Processing Systems (NeurIPS 2022)
We explore the connection between outlierrobust
highdimensional statistics and nonconvex optimization
in the presence of sparsity constraints,
with a focus on the fundamental tasks
of robust sparse mean estimation and robust sparse PCA.
We develop novel and simple optimization formulations for these problems
such that {\em any} approximate stationary point
of the associated optimization problem
yields a nearoptimal solution for the
underlying robust estimation task. As a corollary, we obtain that
any firstorder method that efficiently converges
to stationarity yields an efficient algorithm for these tasks.
The obtained algorithms are simple, practical,
and succeed under broader distributional assumptions
compared to prior work.

Learning General Halfspaces with Adversarial Label Noise via Online Gradient Descent
[abstract]
[proceedings]
I. Diakonikolas, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 39th International Conference on Machine Learning (ICML 2022)
We study the problem of learning general, i.e., not necessarily homogeneous,
halfspaces with adversarial label noise under the Gaussian distribution.
Prior work has provided a sophisticated polynomialtime algorithm for this problem.
In this work, we show that the problem can be solved directly via online gradient descent
applied to a sequence of natural nonconvex surrogates. This approach yields a simple
iterative learning algorithm for general halfspaces with nearoptimal sample complexity,
runtime, and error guarantee. At the conceptual level, our work establishes an intriguing
connection between learning halfspaces with adversarial noise and online optimization
that may find other applications.

Streaming Algorithms for HighDimensional Robust Statistics
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Pensia, T. Pittas
Proceedings of the 39th International Conference on Machine Learning (ICML 2022)
We study highdimensional robust statistics tasks in the streaming model.
A recent line of work obtained computationally efficient algorithms
for a range of highdimensional robust estimation tasks.
Unfortunately, all previous algorithms require storing the entire dataset,
incurring memory at least quadratic in the dimension.
In this work, we develop the first efficient streaming algorithms
for highdimensional robust statistics with nearoptimal
memory requirements (up to logarithmic factors).
Our main result is for the task of highdimensional robust mean estimation
in (a strengthening of) Huber's contamination model.
We give an efficient singlepass
streaming algorithm for this task
with nearoptimal error guarantees and
space complexity nearlylinear in the dimension.
As a corollary, we obtain streaming algorithms with nearoptimal space complexity
for several more complex tasks,
including robust covariance estimation, robust regression,
and more generally robust stochastic optimization.

Learning a Single Neuron with Adversarial Label Noise via Gradient Descent
[abstract]
[arxiv]
I. Diakonikolas, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 35th Annual Conference on Learning Theory (COLT 2022)
We study the fundamental problem of learning a single neuron, i.e., a
function of the form $\x \mapsto \sigma(w \cdot x)$ for monotone activations
$\sigma:\R \mapsto \R$, with respect to the $L_2^2$loss in the presence of adversarial label noise.
Specifically, we are given labeled examples from a distribution $D$ on $(x, y) \in \R^d \times \R$
such that there exists $w^\ast \in \R^d$ achieving $F(w^\ast) = \eps$, where
$F(w) = \E_{(\x,y) \sim D}[(\sigma(w \cdot x)  y)^2]$. The goal of the learner
is to output a hypothesis vector $\wt{w}$ such that $F(\wt{w}) = C \, \eps$ with
high probability, where $C>1$ is a universal constant. As our main contribution, we give
efficient constantfactor approximate learners
for a broad class of distributions (including logconcave distributions)
and activation functions (including ReLUs and sigmoids).
Concretely, for the class of isotropic logconcave distributions, we obtain
the following important corollaries:
For the logistic activation, i.e., $\sigma(t) = 1/(1+e^{t})$, we obtain the first
polynomialtime constant factor approximation (even under the Gaussian distribution).
Our algorithm has sample complexity $\wt{O}(d/\eps)$, which is tight within
polylogarithmic factors.
For the ReLU activation, i.e., $\sigma(t) = \max(0,t)$, we give an efficient algorithm with
sample complexity $\wt{O}(d \, \polylog(1/\eps))$. Prior to our work, the best known
constantfactor approximate learner had sample complexity $\tilde{\Omega}(d/\eps)$.
In both of these settings, our algorithms are simple, performing gradientdescent
on the (regularized) $L_2^2$loss. The correctness of our algorithms relies
on novel structural results that we establish, showing that (essentially all)
stationary points of the underlying nonconvex loss are approximately optimal.

Robust Sparse Mean Estimation via Sum of Squares
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, S. Karmalkar, A. Pensia, T. Pittas
Proceedings of the 35th Annual Conference on Learning Theory (COLT 2022)
We study the problem of highdimensional sparse mean estimation
in the presence of an $\eps$fraction of adversarial outliers.
Prior work obtained sample and computationally efficient algorithms
for this task for identitycovariance subgaussian distributions.
In this work, we develop the first efficient algorithms for robust sparse mean estimation
without a priori knowledge of the covariance.
For distributions on $\R^d$ with ``certifiably bounded''
$t$th moments and sufficiently light tails,
our algorithm achieves error of $O(\eps^{11/t})$ with sample complexity
$m = (k\log(d))^{O(t)}/\eps^{22/t}$.
For the special case of the Gaussian distribution, our algorithm achieves nearoptimal error of
$\tilde O(\eps)$ with sample complexity $m = O(k^4 \polylog(d))/\eps^2$.
Our algorithms follow the SumofSquares based, proofs to algorithms approach.
We complement our upper bounds
with Statistical Query and lowdegree polynomial testing lower bounds,
providing evidence that the sampletimeerror tradeoffs achieved by our algorithms
are qualitatively the best possible.

Optimal SQ Lower Bounds for Robustly Learning Discrete Product Distributions and Ising Models
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, Y. Sun
Proceedings of the 35th Annual Conference on Learning Theory (COLT 2022)
We establish optimal Statistical Query (SQ) lower bounds
for robustly learning certain families of discrete highdimensional distributions.
In particular, we show that no efficient SQ algorithm with access to an $\eps$corrupted
binary product distribution can learn its mean within $\ell_2$error
$o(\eps \sqrt{\log(1/\eps)})$.
Similarly, we show that no efficient SQ algorithm with access to an $\eps$corrupted
ferromagnetic hightemperature Ising model can learn the model
to total variation distance $o(\eps \log(1/\eps))$.
Our SQ lower bounds match the error guarantees of known algorithms for these problems,
providing evidence that current upper bounds for these tasks are best possible.
At the technical level, we develop a generic SQ lower bound for discrete
highdimensional distributions starting from lowdimensional moment matching constructions
that we believe will find other applications. Additionally, we introduce new ideas
to analyze these momentmatching constructions for discrete univariate distributions.

NonGaussian Component Analysis via Lattice Basis Reduction
[abstract]
[arxiv]
I. Diakonikolas, D. Kane
Proceedings of the 35th Annual Conference on Learning Theory (COLT 2022)
NonGaussian Component Analysis (NGCA) is the following distribution learning problem:
Given i.i.d. samples from a distribution on $\R^d$ that is nongaussian in a hidden direction
$v$ and an independent standard Gaussian in the orthogonal directions, the goal is to approximate
the hidden direction $v$. Prior work~\cite{DKS17sq} provided formal evidence
for the existence of an informationcomputation tradeoff for NGCA
under appropriate momentmatching conditions on the univariate nongaussian distribution $A$.
The latter result does not apply when the distribution $A$ is discrete.
A natural question is whether informationcomputation tradeoffs persist in this setting.
In this paper, we answer this question in the negative
by obtaining a sample and computationally efficient algorithm for NGCA
in the regime that $A$ is discrete or nearly discrete, in a welldefined technical sense.
The key tool leveraged in our algorithm is the LLL method~\cite{LLL82}
for lattice basis reduction.

NearOptimal Statistical Query Hardness of Learning Halfspaces with Massart Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane
Proceedings of the 35th Annual Conference on Learning Theory (COLT 2022)
We study the problem of PAC learning halfspaces with Massart noise.
Given labeled samples $(x, y)$
from a distribution $D$ on $\R^{d} \times \{ \pm 1\}$
such that the marginal $D_x$ on the examples is arbitrary
and the label $y$ of example $x$ is generated from the target halfspace
corrupted by a Massart adversary with flipping probability $\eta(x) \leq \eta \leq 1/2$,
the goal is to compute a hypothesis with small misclassification error.
The best known $\poly(d, 1/\eps)$time algorithms for this problem
achieve error of $\eta+\eps$, which can be far from the optimal bound of $\opt+\eps$,
where $\opt = \E_{x \sim D_x} [\eta(x)]$.
While it is known that achieving $\opt+o(1)$ error requires superpolynomial time
in the Statistical Query model, a large gap remains between
known upper and lower bounds.
In this work, we essentially characterize
the efficient learnability of Massart halfspaces in the Statistical Query (SQ) model.
Specifically, we show that no efficient SQ algorithm for learning Massart halfspaces on $\R^d$
can achieve error better than $\Omega(\eta)$, even if $\opt = 2^{\log^{c} (d)}$,
for any universal constant $c \in (0, 1)$. Furthermore, when the noise upper bound $\eta$ is close to $1/2$,
our error lower bound becomes $\eta  o_{\eta}(1)$, where the $o_{\eta}(1)$ term goes to $0$
when $\eta$ approaches $1/2$.
Our results provide strong evidence that known
learning algorithms for Massart halfspaces are nearly best possible,
thereby resolving a longstanding open problem in learning theory.

Learning General Halfspaces with General Massart Noise under the Gaussian Distribution
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 54th Annual ACM Symposium on Theory of Computing (STOC 2022)
We study the problem of PAC learning halfspaces on $\mathbb{R}^d$ with Massart noise under the
Gaussian distribution. In the Massart model, an adversary is allowed to flip the label of each point
$\mathbf{x}$ with unknown probability $\eta(\mathbf{x}) \leq \eta$, for some parameter $\eta \in
[0,1/2]$. The goal is to find a hypothesis with misclassification error of $\mathrm{OPT} + \eps$,
where $\mathrm{OPT}$ is the error of the target halfspace. This problem had been previously studied
under two assumptions: (i) the target halfspace is {\em homogeneous} (i.e., the separating
hyperplane goes through the origin), and (ii) the parameter $\eta$ is {\em strictly} smaller than
$1/2$. Prior to this work, no nontrivial bounds were known when either of these assumptions is
removed. We study the general problem and establish the following:
For $\eta <1/2$, we give a learning algorithm for general halfspaces with sample and
computational complexity $d^{O_{\eta}(\log(1/\gamma))}\mathrm{poly}(1/\epsilon)$, where $\gamma
\coloneqq \max\{\epsilon, \min\{\mathbf{Pr}[f(\mathbf{x}) = 1], \mathbf{Pr}[f(\mathbf{x}) = 1]\}
\}$ is the ``bias'' of the target halfspace $f$. Prior efficient algorithms could only handle the
special case of $\gamma = 1/2$. Interestingly, we establish a qualitatively matching lower bound of
$d^{\Omega(\log(1/\gamma))}$ on the complexity of any Statistical Query (SQ) algorithm.
For $\eta = 1/2$, we give a learning algorithm for general halfspaces with sample and
computational complexity $O_\epsilon(1) \,d^{O(\log(1/\epsilon))}$. This result is new even for the
subclass of homogeneous halfspaces; prior algorithms for homogeneous Massart halfspaces provide
vacuous guarantees for $\eta=1/2$. We complement our upper bound with a nearlymatching SQ lower
bound of $d^{\Omega(\log(1/\epsilon) )}$, which holds even for the special case of homogeneous
halfspaces.
Taken together, our results qualitatively characterize the complexity of learning general halfspaces
with general Massart noise under Gaussian marginals.
Our techniques rely on determining the existence (or nonexistence) of lowdegree polynomials whose
expectations distinguish Massart halfspaces from random noise.

Clustering Mixture Models in AlmostLinear Time via ListDecodable Mean Estimation
[abstract]
[arxiv]
I. Diakonikolas, D. M. Kane, D. Kongsgaard, J. Li, K. Tian
Proceedings of the 54th Annual ACM Symposium on Theory of Computing (STOC 2022)
We study the problem of listdecodable mean estimation, where an adversary can corrupt a majority of the dataset.
Specifically, we are given a set $T$ of $n$ points in $\mathbb{R}^d$ and a parameter $0< \alpha <\frac 1 2$
such that an $\alpha$fraction of the points in $T$ are i.i.d. samples from a wellbehaved distribution
$\mathcal{D}$ and the remaining $(1\alpha)$fraction of the points are arbitrary. The goal is to output
a small list of vectors at least one of which is close to the mean of $\mathcal{D}$. As our main contribution,
we develop new algorithms for listdecodable mean estimation, achieving nearlyoptimal statistical guarantees,
with running time $n^{1 + o(1)} d$. All prior algorithms for this problem had additional polynomial factors
in $\frac 1 \alpha$. As a corollary, we obtain the first {\em almostlinear time} algorithms for clustering mixtures
of $k$ separated wellbehaved distributions, nearlymatching the statistical guarantees of spectral methods.
Prior clustering algorithms inherently relied on an application of $k$PCA, thereby incurring runtimes of
$\Omega(n d k)$. This marks the first runtime improvement for this basic statistical problem in nearly two decades.
The starting point of our approach is a novel and simpler nearlinear time robust mean estimation algorithm
in the $\alpha \to 1$ regime, based on a oneshot matrix multiplicative weightsinspired potential decrease.
We crucially leverage this new algorithmic framework in the context of the iterative multifiltering technique
of \cite{DiakonikolasKS18, DiakonikolasKK20}, providing a method to simultaneously cluster and downsample
points using {\em onedimensional} projections  thus, bypassing the $k$PCA subroutines required by prior algorithms.

Robustly Learning Mixtures of k Arbitrary Gaussians
[abstract]
[arxiv]
A. Bakshi, I. Diakonikolas, H. Jia, D. Kane, P. Kothari, S. Vempala
Proceedings of the 54th Annual ACM Symposium on Theory of Computing (STOC 2022)
We give a polynomialtime algorithm for the problem of robustly estimating a mixture of
$k$ arbitrary Gaussians in $\R^d$, for any fixed $k$, in the presence of a constant fraction of arbitrary corruptions.
This resolves the main open problem in several previous works on algorithmic robust statistics, which addressed
the special cases of robustly estimating (a) a single Gaussian, (b) a mixture of TVdistance separated Gaussians,
and (c) a uniform mixture of two Gaussians. Our main tools are an efficient \emph{partial clustering} algorithm
that relies on the sumofsquares method, and a novel tensor decomposition algorithm that allows errors in both
Frobenius norm and lowrank terms.

Hardness of Learning a Single Neuron with Adversarial Label Noise
[abstract]
[proceedings]
I. Diakonikolas, D. Kane, P. Manurangsi, L. Ren
Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS 2022)
Selected for Oral Presentation
We study the problem of distributionfree learning of a single neuron
under adversarial label noise with respect to the squared loss.
For a wide range of activation functions, including ReLUs and sigmoids, we prove
hardness of learning results in the Statistical Query model and under a wellstudied
assumption on the complexity of refuting XOR formulas. Specifically, we establish
that no polynomialtime learning algorithm, even improper, can approximate
the optimal loss value within any constant factor.

Efficient Approximation Algorithms for the Inverse Semivalue Problem
[abstract]
[proceedings]
I. Diakonikolas, C. Pavlou, J. Peebles, A. Stewart
Proceedings of the 21st International Conference on Autonomous Agents and MultiAgent Systems (AAMAS 2022)
Weighted voting games are typically used to model
situations where a number of agents vote against or for a proposal.
In such games, a proposal is accepted if a weighted sum of the votes
exceeds a specified threshold.
As the influence of a player over the outcome is not
in general proportional to her assigned weight,
various power indices have been proposed to measure each player's influence.
The inverse power index problem is the problem of designing
a weighted voting game that achieves a set of desirable
influences as they are measured by a predefined power index.
Recent work has shown that exactly solving the inverse power index problem is computationally intractable
when the power index is in the class of semivalues  a broad class that includes the popular
Shapley value and Banzhaf index. In this work, we design efficient approximation
algorithms for the inverse semivalue problem. We develop a unified methodology
that leads to computationally efficient algorithms that solve
the inverse semivalue problem to any desired accuracy. We perform an extensive experimental
evaluation of our algorithms on both synthetic and real inputs. Our experiments show
that our algorithms are scalable and achieve higher accuracy
compared to previous methods in the literature.

Forster Decomposition and Learning Halfspaces with Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, C. Tzamos
Advances in Neural Information Processing Systems (NeurIPS 2021)
Selected for Spotlight Presentation
A Forster transform is an operation that turns a distribution into one with good anticoncentration properties.
While a Forster transform does not always exist, we show that any distribution can be efficiently
decomposed as a disjoint mixture of few distributions for which a Forster transform exists
and can be computed efficiently. As the main application of this result, we obtain the first
polynomialtime algorithm for distributionindependent PAC learning of halfspaces
in the Massart noise model with strongly polynomial sample complexity,
i.e., independent of the bit complexity of the examples. Previous algorithms for this learning problem
incurred sample complexity scaling polynomially with the bit complexity,
even though such a dependence is not informationtheoretically necessary.

ReLU Regression with Massart Noise
[abstract]
[arxiv]
I. Diakonikolas, J. Park, C. Tzamos
Advances in Neural Information Processing Systems (NeurIPS 2021)
We study the fundamental problem of ReLU regression, where the goal is
to fit Rectified Linear Units (ReLUs) to data.
This supervised learning task is efficiently solvable in the realizable setting,
but is known to be computationally hard with adversarial label noise.
In this work, we focus on ReLU regression in the Massart noise model,
a natural and wellstudied semirandom noise model. In this model, the label of every point
is generated according to a function in the class, but an adversary is allowed to change
this value arbitrarily with some probability, which is {\em at most} $\eta < 1/2$.
We develop an efficient algorithm that achieves exact parameter recovery in this model
under mild anticoncentration assumptions on the underlying distribution.
Such assumptions are necessary for exact recovery to be informationtheoretically possible.
We demonstrate that our algorithm significantly outperforms naive applications of
$\ell_1$ and $\ell_2$ regression on both synthetic and real data.

Statistical Query Lower Bounds for ListDecodable Linear Regression
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Pensia, T. Pittas, A. Stewart
Advances in Neural Information Processing Systems (NeurIPS 2021)
Selected for Spotlight Presentation
We study the problem of listdecodable linear regression, where an adversary
can corrupt a majority of the examples. Specifically, we are given a set $T$ of labeled examples
$(x, y) \in \R^d \times \R$ and a parameter $0< \alpha <1/2$ such that an $\alpha$fraction
of the points in $T$ are i.i.d.\ samples from a linear regression model with Gaussian covariates,
and the remaining $(1\alpha)$fraction of the points are drawn from an arbitrary noise distribution.
The goal is to output a small list of hypothesis vectors such that at least one of them is close to the target regression vector.
Our main result is a Statistical Query (SQ) lower bound of $d^{\poly(1/\alpha)}$ for this problem.
Our SQ lower bound qualitatively matches the performance of previously developed algorithms,
providing evidence that current upper bounds for this task are nearly best possible.

ListDecodable Mean Estimation in NearlyPCA Time
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, D. Kongsgaard, J. Li, K. Tian
Advances in Neural Information Processing Systems (NeurIPS 2021)
Selected for Spotlight Presentation
Traditionally, robust statistics has focused on designing estimators tolerant to a minority of contaminated data.
Robust {\em listdecodable learning} focuses on the more challenging regime
where only a minority $1/k$ fraction of the dataset is drawn from the distribution of interest,
for some $k \ge 2$, and no assumptions are made on the remaining data. In this paper, we study the fundamental task
of listdecodable mean estimation in high dimensions. Our main result is a new listdecodable mean estimation algorithm
for bounded covariance distributions with optimal sample complexity and error rate, running in {\em nearlyPCA time}.
Specifically, assuming the ground truth distribution on $\R^d$ has covariance bounded by the identity,
our algorithm outputs a list of $O(k)$ candidate means, one of which is within distance $O(\sqrt{k})$ from the true mean.
Our algorithm runs in time $\widetilde{O}(ndk)$ for all $k = O(\sqrt{d}) \cup \Omega(d)$, where $n$ is the size of the dataset.
We also show that a variant of our algorithm has runtime $\widetilde{O}(ndk)$ for all $k$, at the expense of an $O(\sqrt{\log k})$
factor in the recovery guarantee. This runtime matches up to logarithmic factors the cost of performing a single $k$PCA on the data,
which is a natural bottleneck of known algorithms for (very) special cases of our problem, such as clustering wellseparated mixtures.
Prior to our work, the fastest listdecodable mean estimation algorithms had runtimes $\widetilde{O}(n^2 d k^2)$~\cite{DiakonikolasKK20},
and $\widetilde{O}(nd k^C)$ \cite{CherapanamjeriMY20} for an unspecified constant $C \geq 6$.
Our approach builds on a novel soft downweighting method, which is arguably the simplest known polynomialtime mean estimation technique
in the listdecodable learning setting. To develop our fast algorithms, we boost the computational cost of our simpler algorithm
via a careful ``winwinwin'' analysis of an approximate Ky Fan matrix multiplicative weights procedure we develop,
which we believe may be of independent interest.

Agnostic Proper Learning of Halfspaces under Gaussian Marginals
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 34th Annual Conference on Learning Theory (COLT 2021)
We study the problem of agnostically learning halfspaces under the Gaussian distribution.
Our main result is the {\em first proper} learning algorithm for this problem whose
sample complexity and computational complexity qualitatively match those of the best known
improper agnostic learner. Building on this result, we also obtain the first proper
polynomialtime approximation scheme (PTAS) for agnostically learning homogeneous halfspaces.
Our techniques naturally extend to agnostically learning linear models with respect
to other nonlinear activations, yielding in particular the first proper agnostic algorithm
for ReLU regression.

The Optimality of Polynomial Regression for Agnostic Learning under Gaussian Marginals
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, T. Pittas, N. Zarifis
Proceedings of the 34th Annual Conference on Learning Theory (COLT 2021)
We study the problem of agnostic learning under the Gaussian distribution.
We develop a method for finding hard families of examples for a wide class of problems
by using LP duality. For Booleanvalued concept classes, we show that the $L^1$regression algorithm
is essentially best possible, and therefore that the computational difficulty of agnostically
learning a concept class is closely related to the polynomial degree required to approximate
any function from the class in $L^1$norm. Using this characterization along with additional
analytic tools, we obtain optimal SQ lower bounds for agnostically learning linear threshold
functions and the first nontrivial SQ lower bounds for polynomial threshold functions
and intersections of halfspaces. We also develop an analogous theory for agnostically learning
realvalued functions, and as an application prove nearoptimal SQ lower bounds for
agnostically learning ReLUs and sigmoids.

Boosting in the Presence of Massart Noise
[abstract]
[arxiv]
I. Diakonikolas, R. Impagliazzo, D. M. Kane, R. Lei, J. Sorrell, C. Tzamos
Proceedings of the 34th Annual Conference on Learning Theory (COLT 2021)
We study the problem of boosting the accuracy of a weak learner in the (distributionindependent) PAC model with Massart noise.
In the Massart noise model, the label of each example $x$ is independently misclassified with probability $\eta(x) \leq \eta$, where $\eta<1/2$.
The Massart model lies between the random classification noise model and the agnostic model.
Our main positive result is the first computationally efficient boosting algorithm in the presence of Massart noise
that achieves misclassification error arbitrarily close to $\eta<1/2$.
Prior to our work, no nontrivial booster was known in this setting. Moreover, we show that this error upper bound is best possible
for polynomialtime blackbox boosters, under standard cryptographic assumptions.
Our upper and lower bounds characterize the complexity of boosting in the distributionindependent PAC model with Massart noise.
As a simple application of our positive result, we give the first efficient Massart learner for unions of highdimensional rectangles.

OutlierRobust Learning of Ising Models Under Dobrushin's Condition
[abstract]
[arxiv]
I. Diakonikolas, D. M. Kane, A. Stewart, Y. Sun
Proceedings of the 34th Annual Conference on Learning Theory (COLT 2021)
We study the problem of learning Ising models satisfying Dobrushin's
condition in the outlierrobust setting where a constant fraction of
the samples are adversarially corrupted. Our main result is to provide
the first computationally efficient robust learning algorithm for this
problem with nearoptimal error guarantees. Our algorithm can be seen
as a special case of an algorithm for robustly learning a distribution
from a general exponential family. To prove its correctness for
Ising models, we establish new anticoncentration results
for degree$2$ polynomials of Ising models
that may be of independent interest.

The Sample Complexity of Robust Covariance Testing
[abstract]
[arxiv]
I. Diakonikolas, D. Kane
Proceedings of the 34th Annual Conference on Learning Theory (COLT 2021)
We study the problem of testing the covariance matrix of a highdimensional Gaussian
in a robust setting, where the input distribution has been corrupted in Huber's contamination model.
Specifically, we are given i.i.d. samples from a distribution of the form $Z = (1\eps) X + \eps B$,
where $X$ is a zeromean and unknown covariance Gaussian $\mathcal{N}(0, \Sigma)$,
$B$ is a fixed but unknown noise distribution, and $\eps>0$ is an arbitrarily small constant representing
the proportion of contamination.
We want to distinguish between the cases that $\Sigma$ is the identity matrix versus $\gamma$far from the identity
in Frobenius norm.
In the absence of contamination, prior work gave a simple tester
for this hypothesis testing task that uses $O(d)$ samples. Moreover, this sample upper bound was shown
to be best possible, within constant factors. Our main result is that the sample complexity of covariance testing
dramatically increases in the contaminated setting. In particular, we prove a sample complexity lower bound
of $\Omega(d^2)$ for $\eps$ an arbitrarily small constant and $\gamma = 1/2$.
This lower bound is best possible, as $O(d^2)$ samples suffice to even robustly {\em learn}
the covariance. The conceptual implication of our result is that, for the natural setting we consider,
robust hypothesis testing is at least as hard as robust estimation.

Learning Online Algorithms with Distributional Advice
I. Diakonikolas, V. Kontonis, C. Tzamos, A. Vakilian, N. Zarifis
Proceedings of the 38th International Conference on Machine Learning (ICML 2021)

A Polynomial Time Algorithm for Learning Halfspaces with Tsybakov Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 53rd Annual ACM Symposium on Theory of Computing (STOC 2021)
We study the problem of PAC learning homogeneous halfspaces in the presence of Tsybakov noise.
In the Tsybakov noise model, the label of every sample is independently flipped with an adversarially
controlled probability that can be arbitrarily close to $1/2$ for a fraction of the samples.
{\em We give the first polynomialtime algorithm for this fundamental learning problem.}
Our algorithm learns the true halfspace within any desired accuracy $\eps$ and
succeeds under a broad family of wellbehaved distributions including logconcave distributions.
Prior to our work, the only previous algorithm for this problem
required quasipolynomial runtime in $1/\eps$.
Our algorithm employs a recently developed reduction~\cite{DKTZ20b} from learning
to certifying the nonoptimality of a candidate halfspace. This prior work
developed a quasipolynomial time certificate algorithm based on polynomial regression.
{\em The main technical contribution of the current paper is the first polynomialtime certificate algorithm.}
Starting from a nontrivial warmstart, our algorithm performs
a novel ``winwin'' iterative process which, at each step, either finds a valid certificate
or improves the angle between the current halfspace and the true one.
Our warmstart algorithm for isotropic logconcave distributions involves
a number of analytic tools that may be of broader interest. These include
a new efficient method for reweighting the distribution in order to recenter it
and a novel characterization of the spectrum of the degree$2$ Chow parameters.

Learning Halfspaces with Tsybakov Noise
[abstract]
[arxiv]
I. Diakonikolas, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 53rd Annual ACM Symposium on Theory of Computing (STOC 2021)
(Conference version to be merged with paper above.)
We study the efficient PAC learnability of halfspaces in the presence of Tsybakov noise.
In the Tsybakov noise model, each label is independently flipped with some probability
which is controlled by an adversary. This noise model significantly generalizes the Massart noise model,
by allowing the flipping probabilities to be arbitrarily close to $1/2$ for a fraction of the samples.
Our main result is the first nontrivial PAC learning algorithm for this problem under
a broad family of structured distributions  satisfying certain concentration
and (anti)anticoncentration properties  including logconcave distributions.
Specifically, we given an algorithm that achieves misclassification error $\eps$
with respect to the true halfspace, with quasipolynomial runtime dependence in $1/\eps$.
The only previous upper bound for this problem  even for the special case of logconcave distributions 
was doubly exponential in $1/\eps$ (and follows via the naive reduction to agnostic learning).
Our approach relies on a novel computationally efficient procedure to certify whether a candidate solution is nearoptimal,
based on semidefinite programming. We use this certificate procedure as a blackbox and turn it
into an efficient learning algorithm by searching over the space of halfspaces
via online convex optimization.

Optimal Testing of Discrete Distributions with High Probability
[abstract]
[arxiv]
I. Diakonikolas, T. Gouleakis, D. Kane, J. Peebles, E. Price
Proceedings of the 53rd Annual ACM Symposium on Theory of Computing (STOC 2021)
We study the problem of testing discrete distributions with a focus on the high probability regime.
Specifically, given samples from one or more discrete distributions, a property $\mathcal{P}$, and
parameters $0< \eps, \delta <1$, we want to distinguish {\em with probability at least $1\delta$}
whether these distributions satisfy $\mathcal{P}$ or are $\eps$far from $\mathcal{P}$
in total variation distance. Most prior work in distribution testing studied the constant confidence case
(corresponding to $\delta = \Omega(1)$), and provided sampleoptimal testers for a range of properties.
While one can always boost the confidence probability of any such tester by blackbox amplification,
this generic boosting method typically leads to suboptimal sample bounds.
Here we study the following broad question: For a given property $\mathcal{P}$, can we {\em characterize}
the sample complexity of testing $\mathcal{P}$ as a function of all relevant problem parameters,
including the error probability $\delta$? Prior to this work, uniformity testing was the only statistical task
whose sample complexity had been characterized in this setting. As our main results,
we provide the first algorithms for closeness and independence testing that are sampleoptimal, within
constant factors, as a function of all relevant parameters. We also show matching
informationtheoretic lower bounds on the sample complexity of these problems.
Our techniques naturally extend to give optimal testers for related problems.
To illustrate the generality of our methods, we give optimal algorithms for testing collections of distributions
and testing closeness with unequal sized samples.

Rapid Approximate Aggregation with DistributionSensitive Interval Guarantees
[abstract]
[arxiv]
S. Macke, M. Aliakbarpour, I. Diakonikolas, A. Parameswaran, R. Rubinfeld
Proceedings of the 37th IEEE International Conference on Data Engineering (ICDE 2021)
Aggregating data is fundamental to data analytics, data exploration, and OLAP.
Approximate query processing (AQP) techniques are often used to accelerate computation of aggregates using samples,
for which confidence intervals (CIs) are widely used to quantify the associated error. CIs used in practice fall
into two categories: techniques that are tight but not correct, i.e., they yield tight intervals but only offer asymptotic guarantees,
making them unreliable, or techniques that are correct but not tight, i.e., they offer rigorous guarantees,
but are overly conservative, leading to confidence intervals that are too loose to be useful.
In this paper, we develop a CI technique that is both correct and tighter than traditional approaches.
Starting from conservative CIs, we identify two issues they often face: pessimistic mass allocation (PMA) and
phantom outlier sensitivity (PHOS). By developing a novel rangetrimming technique for eliminating PHOS
and pairing it with known CI techniques without PMA, we develop a technique for computing CIs with strong guarantees
that requires fewer samples for the same width. We implement our techniques underneath a samplingoptimized inmemory
column store and show how to accelerate queries involving aggregates on a real dataset with speedups of up to 124x
over traditional AQPwithguarantees and more than 1000x over exact methods.

Outlier Robust Mean Estimation with Subgaussian Rates via Stability
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Pensia
Advances in Neural Information Processing Systems (NeurIPS 2020)
We study the problem of outlier robust highdimensional mean estimation under a finite covariance assumption,
and more broadly under finite lowdegree moment assumptions. We consider a standard stability condition
from the recent robust statistics literature and prove that, except with exponentially small failure probability,
there exists a large fraction of the inliers satisfying this condition. As a corollary, it follows that a number
of recently developed algorithms for robust mean estimation, including iterative filtering and nonconvex gradient descent,
give optimal error estimators with (near)subgaussian rates. Previous analyses of these algorithms gave significantly suboptimal rates.
As a corollary of our approach, we obtain the first computationally efficient algorithm with subgaussian rate
for outlierrobust mean estimation in the strong contamination model under a finite covariance assumption.

The Complexity of Adversarially Robust Proper Learning of Halfspaces with Agnostic Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, P. Manurangsi
Advances in Neural Information Processing Systems (NeurIPS 2020)
We study the computational complexity of adversarially robust proper learning of halfspaces in the distributionindependent
agnostic PAC model, with a focus on $L_p$ perturbations. We give a computationally efficient learning algorithm
and a nearly matching computational hardness result for this problem. An interesting implication of our findings is that
the $L_{\infty}$ perturbations case is provably computationally harder than the case $2 \leq p < \infty$.

NearOptimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, N. Zarifis
Advances in Neural Information Processing Systems (NeurIPS 2020)
We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals.
In the former problem, given labeled examples $(\bx, y)$ from an unknown distribution on $\R^d \times \{ \pm 1\}$,
whose marginal distribution on $\bx$ is the standard Gaussian and the labels $y$ can be arbitrary,
the goal is to output a hypothesis with 01 loss $\opt+\eps$, where $\opt$
is the 01 loss of the bestfitting halfspace.
In the latter problem, given labeled examples $(\bx, y)$ from an unknown distribution on $\R^d \times \R$,
whose marginal distribution on $\bx$ is the standard Gaussian and the labels $y$ can be arbitrary,
the goal is to output a hypothesis with square loss $\opt+\eps$, where $\opt$
is the square loss of the bestfitting ReLU.
We prove Statistical Query (SQ) lower bounds of $d^{\poly(1/\eps)}$ for both of these problems.
Our SQ lower bounds provide strong evidence that current upper bounds for these tasks
are essentially best possible.

ListDecodable Mean Estimation via Iterative MultiFiltering
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, D. Kongsgaard
Advances in Neural Information Processing Systems (NeurIPS 2020)
We study the problem of {\em listdecodable mean estimation} for bounded covariance distributions.
Specifically, we are given a set T of points in with the promise that an unknown $\alpha$fraction of points in T,
where $0<\alpha<1/2$, are drawn from an unknown mean and bounded covariance distribution D, and no assumptions
are made on the remaining points. The goal is to output a small list of hypothesis vectors such that at least one of them
is close to the mean of D. We give the first practically viable estimator for this problem. In more detail, our algorithm
is sample and computationally efficient, and achieves informationtheoretically nearoptimal error. While the only prior
algorithm for this setting inherently relied on the ellipsoid method, our algorithm is iterative and only uses spectral techniques.
Our main technical innovation is the design of a soft outlier removal procedure for highdimensional heavytailed datasets
with a majority of outliers.

NonConvex SGD Learns Halfspaces with Adversarial Label Noise
[abstract]
[arxiv]
I. Diakonikolas, V. Kontonis, C. Tzamos, N. Zarifis
Advances in Neural Information Processing Systems (NeurIPS 2020)
We study the problem of agnostically learning homogeneous halfspaces in the distributionspecific PAC model.
For a broad family of structured distributions, including logconcave distributions, we show that nonconvex
SGD efficiently converges to a solution with misclassification error $O(\opt)+\eps$, where $\opt$ is the
misclassification error of the bestfitting halfspace. In sharp contrast, we show that optimizing
any convex surrogate inherently leads to misclassification error of $\omega(\opt)$, even under Gaussian marginals.

Small Covers for NearZero Sets of Polynomials and Learning Latent Variable Models
[abstract]
[arxiv]
I. Diakonikolas, D. Kane
Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2020)
Let $V$ be any vector space of multivariate degree$d$ homogeneous polynomials with codimension at most $k$,
and $S$ be the set of points where all polynomials in $V$ {\em nearly} vanish. We establish a qualitatively
optimal upper bound on the size of $\eps$covers for $S$, in the $\ell_2$norm. Roughly speaking, we show
that there exists an $\eps$cover for $S$ of cardinality $M = (k/\eps)^{O_d(k^{1/d})}$.
Our result is constructive yielding an algorithm to compute such an $\eps$cover that runs in time $\poly(M)$.
Building on our structural result,
we obtain significantly improved learning algorithms for several fundamental highdimensional
probabilistic models with hidden variables. These include density and parameter estimation for $k$mixtures
of spherical Gaussians (with known common covariance),
PAC learning onehiddenlayer ReLU networks with $k$ hidden units (under the Gaussian distribution),
density and parameter estimation for $k$mixtures of linear regressions (with Gaussian covariates), and
parameter estimation for $k$mixtures of hyperplanes. Our algorithms run in time {\em quasipolynomial}
in the parameter $k$. Previous algorithms for these problems had running times exponential in $k^{\Omega(1)}$.

Robustly Learning any Clusterable Mixture of Gaussians
[abstract]
[arxiv]
I. Diakonikolas, S. Hopkins, D. Kane, S. Karmalkar
Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2020)
(Conference version to be merged with this paper)
We study the efficient learnability of highdimensional Gaussian mixtures in the outlierrobust setting,
where a small constant fraction of the data is adversarially corrupted. We resolve the polynomial learnability of this problem
when the components are pairwise separated in total variation distance. Specifically, we provide an algorithm that,
for any constant number of components $k$, runs in polynomial time and learns the components of an $\eps$corrupted $k$mixture
within information theoretically nearoptimal error of $\tilde{O}(\eps)$, under the assumption that the overlap between any pair of
components $P_i, P_j$ (i.e., the quantity $1TV(P_i, P_j)$) is bounded by $\mathrm{poly}(\eps)$.
Our separation condition is the qualitatively weakest assumption under which accurate clustering of the samples is possible.
In particular, it allows for components with arbitrary covariances and for components with identical means,
as long as their covariances differ sufficiently.
Ours is the first polynomial time algorithm for this problem, even for $k=2$.
Our algorithm follows the SumofSquares based \emph{proofs to algorithms} approach.
Our main technical contribution is a new robust identifiability proof of clusters from a Gaussian mixture,
which can be captured by the constantdegree Sum of Squares proof system. The key ingredients of this proof
are a novel use of \emph{SoScertifiable anticoncentration} and a new characterization of pairs of Gaussians
with small (dimensionindependent) overlap in terms of their parameter distance.

HighDimensional Robust Mean Estimation via Gradient Descent
[abstract]
[arxiv]
Y. Cheng, I. Diakonikolas, R. Ge, M. Soltanolkotabi
Proceedings of the 37th International Conference on Machine Learning (ICML 2020)
We study the problem of highdimensional robust mean estimation
in the presence of a constant fraction of adversarial outliers.
A recent line of work has provided sophisticated polynomialtime algorithms
for this problem with dimensionindependent error guarantees for a range
of natural distribution families.
In this work, we show that a natural nonconvex formulation of the problem
can be solved directly by gradient descent. Our approach leverages a novel structural lemma,
roughly showing that any approximate stationary point of our nonconvex objective
gives a nearoptimal solution to the underlying robust estimation task.
Our work establishes an intriguing connection between algorithmic highdimensional
robust statistics and nonconvex optimization, which may have broader applications
to other robust estimation tasks.

Efficiently Learning Adversarially Robust Halfspaces with Noise
[abstract]
[arxiv]
O. Montasser, S. Goel, I. Diakonikolas, N. Srebro
Proceedings of the 37th International Conference on Machine Learning (ICML 2020)
We study the problem of learning adversarially robust halfspaces
in the distributionindependent setting. In the realizable setting,
we provide necessary and sufficient conditions on the adversarial perturbation sets
under which halfspaces are efficiently robustly learnable. In the presence of random label noise,
we give a simple computationally efficient algorithm for this problem with respect to any
$\ell_p$perturbation.

Efficient Algorithms for Multidimensional Segmented Regression
[abstract]
[arxiv]
I. Diakonikolas, J. Li, A. Voloshinov
Manuscript, 2020
We study the fundamental problem of fixed design {\em multidimensional segmented regression}:
Given noisy samples from a function f, promised to be piecewise linear on an unknown set of k rectangles,
we want to recover f up to a desired accuracy in meansquared error. We provide the first sample and
computationally efficient algorithm for this problem in any fixed dimension. Our algorithm relies on
a simple iterative merging approach, which is novel in the multidimensional setting. Our experimental
evaluation on both synthetic and real datasets shows that our algorithm is competitive and in some cases
outperforms stateoftheart heuristics.

Algorithms and SQ Lower Bounds for PAC Learning OneHiddenLayer ReLU Networks
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Kontonis, N. Zarifis
Proceedings of the 33rd Annual Conference on Learning Theory (COLT 2020)
We study the problem of PAC learning onehiddenlayer ReLU networks
with $k$ hidden units on $\R^d$ under Gaussian marginals in the presence of additive label noise.
For the case of positive coefficients, we give the first polynomialtime algorithm
for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$.
Previously, no polynomial time algorithm was known, even for $k=3$.
This answers an open question posed by Klivans (2017). Importantly,
our algorithm does not require any assumptions about the rank of the weight matrix
and its complexity is independent of its condition number. On the negative side,
for the more general task of PAC learning onehiddenlayer ReLU networks
with positive or negative coefficients, we prove a Statistical Query lower
bound of $d^{\Omega(k)}$. Thus, we provide a separation between the two classes
in terms of efficient learnability. Our upper and lower bounds are general,
extending to broader families of activation functions.

Approximation Schemes for ReLU Regression
[abstract]
[arxiv]
I. Diakonikolas, S. Goel, S. Karmalkar, A. Klivans, M. Soltanolkotabi
Proceedings of the 33rd Annual Conference on Learning Theory (COLT 2020)
We consider the fundamental problem of ReLU regression, where the goal is to output
the best fitting ReLU with respect to square loss given access to draws from some unknown distribution.
We give the first efficient, constantfactor approximation algorithm for this problem assuming
the underlying distribution satisfies some weak concentration and anticoncentration conditions
(and includes, for example, all logconcave distributions). This solves the main open problem of Goel et al.,
who proved hardness results for any exact algorithm for ReLU regression (up to an additive $\epsilon$).
Using more sophisticated techniques, we can improve our results and obtain a polynomialtime approximation
scheme for any subgaussian distribution. Given the aforementioned hardness results, these guarantees
can not be substantially improved.
Our main insight is a new characterization of {\em surrogate losses} for nonconvex activations.
While prior work had established the existence of convex surrogates for monotone activations,
we show that properties of the underlying distribution actually induce strong convexity for the loss,
allowing us to relate the global minimum to the activation's {\em Chow parameters.}

Learning Halfspaces with Massart Noise Under Structured Distributions
[abstract]
[arxiv]
I. Diakonikolas, V. Kontonis, C. Tzamos, N. Zarifis
Proceedings of the 33rd Annual Conference on Learning Theory (COLT 2020)
We study the problem of learning halfspaces with Massart noise in the distributionspecific
PAC model. We give the first computationally efficient algorithm for this problem
with respect to a broad family of distributions, including logconcave distributions.
This resolves an open question posed in a number of prior works. Our approach is extremely
simple: We identify a smooth {\em nonconvex} surrogate loss with the property that any
approximate stationary point of this loss defines a halfspace that is close to the
target halfspace. Given this structural result, we can use SGD to solve the underlying
learning problem.

Private Testing of Distributions via Sample Permutations
[abstract]
[proceedings]
M. Aliakbarpour, I. Diakonikolas, D. Kane, R. Rubinfeld
Advances in Neural Information Processing Systems (NeurIPS 2019)
Statistical tests are at the heart of many scientific tasks. To validate their hypothesis,
researchers in medical and social sciences use individuals' data. The sensitivity of participants'
data requires the design of statistical tests that ensure the privacy of the individuals in the most efficient way.
In this paper, we use the framework of property testing to design algorithms to test the properties
of the distribution that the data is drawn from with respect to differential privacy. In particular,
we investigate testing two fundamental properties of distributions:
(1) testing the equivalence of two distributions when we have unequal numbers of samples from the two distributions.
(2) Testing independence of two random variables. In both cases, we show that our testers achieve near optimal
sample complexity (up to logarithmic factors). Moreover, our dependence on the privacy parameter
is an additive term, which indicates that differential privacy can be obtained in most regimes
of parameters for free.

Nearly Tight Bounds for Robust Proper Learning of Halfspaces with a Margin
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, P. Manurangsi
Advances in Neural Information Processing Systems (NeurIPS 2019)
Selected for Spotlight Presentation at NeurIPS 2019
We study the problem of {\em properly} learning large margin halfspaces in the agnostic PAC model.
In more detail, we study the complexity of properly learning $d$dimensional halfspaces
on the unit ball within misclassification error $\alpha \cdot \mathrm{OPT}_{\gamma} + \epsilon$,
where $\mathrm{OPT}_{\gamma}$ is the optimal $\gamma$margin error rate
and $\alpha \geq 1$ is the approximation ratio.
We give learning algorithms and computational hardness results
for this problem, for all values of the approximation ratio $\alpha \geq 1$,
that are nearlymatching for a range of parameters.
Specifically, for the natural setting that $\alpha$ is any constant bigger
than one, we provide an essentially tight complexity characterization.
On the positive side, we give an $\alpha = 1.01$approximate proper learner
that uses $O(1/(\epsilon^2\gamma^2))$ samples (which is optimal) and runs in time
$\mathrm{poly}(d/\epsilon) \cdot 2^{\tilde{O}(1/\gamma^2)}$. On the negative side,
we show that {\em any} constant factor approximate proper learner has runtime
$\mathrm{poly}(d/\epsilon) \cdot 2^{(1/\gamma)^{2o(1)}}$,
assuming the Exponential Time Hypothesis.

OutlierRobust HighDimensional Sparse Estimation via Iterative Filtering
[abstract]
[arxiv]
I. Diakonikolas, S. Karmalkar, D. Kane, E. Price, A. Stewart
Advances in Neural Information Processing Systems (NeurIPS 2019)
We study highdimensional sparse estimation tasks in a robust setting where a constant fraction
of the dataset is adversarially corrupted. Specifically, we focus on the fundamental problems of robust
sparse mean estimation and robust sparse PCA.
We give the first practically viable robust estimators for these problems.
In more detail, our algorithms are sample and computationally efficient
and achieve nearoptimal robustness guarantees.
In contrast to prior provable algorithms which relied on the ellipsoid method,
our algorithms use spectral techniques to iteratively remove outliers from the dataset.
Our experimental evaluation on synthetic data shows that our algorithms are scalable and
significantly outperform a range of previous approaches, nearly matching the best error rate without corruptions.

DistributionIndependent PAC Learning of Halfspaces with Massart Noise
[abstract]
[arxiv]
I. Diakonikolas, T. Gouleakis, C. Tzamos
Advances in Neural Information Processing Systems (NeurIPS 2019)
Outstanding Paper Award at NeurIPS 2019
See here
for a description of our work by the program chairs
and here for the slides of my NeurIPS talk.
We study the problem of {\em distributionindependent} PAC learning of halfspaces in the presence of Massart noise.
Specifically, we are given a set of labeled examples $(\bx, y)$ drawn
from a distribution $\D$ on $\R^{d+1}$ such that the marginal distribution
on the unlabeled points $\bx$ is arbitrary and the labels $y$ are generated by an unknown halfspace
corrupted with Massart noise at noise rate $\eta<1/2$. The goal is to find
a hypothesis $h$ that minimizes the misclassification error $\pr_{(\bx, y) \sim \D} \left[ h(\bx) \neq y \right]$.
We give a $\poly(d, 1/\eps)$ time algorithm for this problem with misclassification error $\eta+\eps$.
We also provide evidence that improving on the error guarantee of our algorithm
might be computationally hard. Prior to our work, no efficient weak (distributionindependent) learner
was known in this model, even for the class of disjunctions. The existence of such an algorithm
for halfspaces (or even disjunctions) has been posed as an open question in various works,
starting with Sloan (1988), Cohen (1997), and was most recently highlighted in Avrim Blum's FOCS 2003 tutorial.

Equipping Experts/Bandits with Longterm Memory
[abstract]
[arxiv]
K. Zheng, H. Luo, I. Diakonikolas, L. Wang
Advances in Neural Information Processing Systems (NeurIPS 2019)
We propose the first reductionbased approach to obtaining longterm memory guarantees for online learning
in the sense of Bousquet and Warmuth, 2002,by reducing the problem to achieving typical switching regret.
Specifically, for the classical expert problem with $K$ actions and $T$ rounds, using our framework
we develop various algorithms with a regret bound of order $O(\sqrt{T(S\ln T + n \ln K)})$ compared
to any sequence of experts with $S1$ switches among $n \leq \min\{S, K\}$ distinct experts.
In addition, by plugging specific adaptive algorithms into our framework, we also achieve the best of both
stochastic and adversarial environments simultaneously.
This resolves an open problem of Warmuth and Koolen, 2014.
Furthermore, we extend our results to the sparse multiarmed bandit setting and show both negative
and positive results for longterm memory guarantees.
As a side result, our lower bound also implies that sparse losses do not help improve
the worstcase regret for contextual bandits, a sharp contrast with the noncontextual case.

A Polynomial Time Algorithm for LogConcave Maximum Likelihood via Locally Exponential Families
[abstract]
[arxiv]
B. Axelrod, I. Diakonikolas, A. Sidiropoulos, A. Stewart, G. Valiant
Advances in Neural Information Processing Systems (NeurIPS 2019)
We study the problem of computing the maximum likelihood estimator (MLE) of multivariate logconcave densities.
Our main result is the first computationally efficient algorithm for this problem. In more detail,
we give an algorithm that, on input a set of $n$ points in $\R^d$ and an accuracy parameter $\epsilon>0$,
it runs in time $\poly(n, d, 1/\epsilon)$, and outputs a logconcave density that with high probability
maximizes the loglikelihood up to an additive $\epsilon$. Our approach relies on a natural convex optimization
formulation of the underlying problem that can be efficiently solved by a projected stochastic subgradient method.
The main challenge lies in showing that a stochastic subgradient of our objective function can be efficiently approximated.
To achieve this, we rely on structural results on approximation of logconcave densities and
leverage classical algorithmic tools on volume approximation of convex bodies and uniform sampling
from convex sets.

Faster Algorithms for HighDimensional Robust Covariance Estimation
[abstract]
[arxiv]
Y. Cheng, I. Diakonikolas, R. Ge, D. Woodruff
Proceedings of the 32nd Annual Conference on Learning Theory (COLT 2019)
We study the problem of estimating the covariance matrix of a highdimensional distribution
when a small constant fraction of the samples can be arbitrarily corrupted.
Recent work gave the first polynomial time algorithms for this problem with nearoptimal
error guarantees for several natural structured distributions.
Our main contribution is to develop faster algorithms for this problem whose running time
nearly matches that of computing the empirical covariance.
Given $N = \tilde{\Omega}(d^2/\epsilon^2)$ samples from a $d$dimensional Gaussian distribution,
an $\epsilon$fraction of which may be arbitrarily corrupted, our algorithm runs in time
$\tilde{O}(d^{3.26})/\poly(\epsilon)$ and approximates the unknown covariance matrix
to optimal error up to a logarithmic factor. Previous robust algorithms with comparable error guarantees
all have runtimes $\tilde{\Omega}(d^{2 \omega})$ when $\epsilon = \Omega(1)$, where $\omega$ is the
exponent of matrix multiplication. We also provide evidence that improving the running time of our
algorithm may require new algorithmic techniques.

Communication and Memory Efficient Testing of Discrete Distributions
[abstract]
[arxiv]
I. Diakonikolas, T. Gouleakis, D. Kane, S. Rao
Proceedings of the 32nd Annual Conference on Learning Theory (COLT 2019)
We study distribution testing with communication and memory constraints
in the following computational models: (1) The {\em onepass streaming model}
where the goal is to minimize the sample complexity of the protocol subject to a memory constraint,
and (2) A {\em distributed model} where the data samples reside at multiple machines and the
goal is to minimize the communication cost of the protocol. In both these models, we provide efficient
algorithms for uniformity/identity testing (goodness of fit) and closeness testing (two sample testing).
Moreover, we show nearlytight lower bounds on (1) the sample complexity
of any onepass streaming tester for uniformity, subject to the memory constraint,
and (2) the communication cost of any uniformity testing protocol,
in a restricted ``onepass'' model of communication.

Testing Identity of Multidimensional Histograms
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, J. Peebles
Proceedings of the 32nd Annual Conference on Learning Theory (COLT 2019)
We investigate the problem of identity testing for multidimensional histogram distributions.
A distribution $p: D \to \R_+$, where $D \subseteq \R^d$, is called a $k$histogram if there exists a partition of the
domain into $k$ axisaligned rectangles such that $p$ is constant within each such rectangle.
Histograms are one of the most fundamental nonparametric families of distributions and
have been extensively studied in computer science and statistics.
We give the first identity tester for this problem with {\em sublearning} sample complexity
in any fixed dimension and a nearlymatching sample complexity lower bound.
More specifically, let $q$ be an unknown $d$dimensional $k$histogram and $p$ be an explicitly given $k$histogram.
We want to correctly distinguish, with probability at least $2/3$, between the case that $p = q$ versus $\pq\_1 \geq \eps$.
We design a computationally efficient algorithm for this hypothesis testing problem
with sample complexity $O((\sqrt{k}/\eps^2) \log^{O(d)}(k/\eps))$. Our algorithm is robust to model misspecification, i.e.,
succeeds even if $q$ is only promised to be {\em close} to a $k$histogram.
Moreover, for $k = 2^{\Omega(d)}$, we show a nearlymatching sample complexity lower bound of
$\Omega((\sqrt{k}/\eps^2) (\log(k/\eps)/d)^{\Omega(d)})$ when $d\geq 2$.
Prior to our work, the sample complexity of the $d=1$ case was wellunderstood,
but no algorithm with sublearning sample complexity was known, even for $d=2$. Our new upper and lower bounds
have interesting conceptual implications regarding the relation between learning and testing in this setting.

Sever: A Robust MetaAlgorithm for Stochastic Optimization
[abstract]
[arxiv]
[code]
I. Diakonikolas, G. Kamath, D. Kane, J. Li, J. Steinhardt, A. Stewart
Proceedings of the 36th International Conference on Machine Learning (ICML 2019)
Oral Presentation at the NeurIPS 2018 Workshop on Security in Machine Learning (SECML 2018)
In high dimensions, most machine learning methods are brittle to even a small
fraction of structured outliers. To address this, we introduce a new
metaalgorithm that can take in a \emph{base learner} such as least squares or stochastic
gradient descent, and harden the learner to be resistant to outliers.
Our method, Sever, possesses strong theoretical guarantees yet is also highly scalablebeyond
running the base learner itself, it only requires computing the top singular vector of a certain
$n \times d$ matrix.
We apply Sever on a drug design dataset and a spam classification dataset, and
find that in both cases it has substantially greater robustness than several baselines.
On the spam dataset, with $1\%$ corruptions, we achieved $7.4\%$ test error,
compared to $13.4\%20.5\%$ for the baselines, and $3\%$ error on the uncorrupted dataset.
Similarly, on the drug design dataset, with $10\%$ corruptions, we achieved $1.42$ meansquared
test error, compared to $1.51$$2.33$ for the baselines, and $1.23$ error on the uncorrupted dataset.

Degreed Chow Parameters Robustly Determine Degreed PTFs (and Algorithmic Applications)
[abstract]
[eccc]
I. Diakonikolas, D. Kane
Proceedings of the 51st Annual ACM Symposium on Theory of Computing (STOC 2019)
The degree$d$ Chow parameters of a Boolean function are its degree at most $d$ Fourier coefficients.
It is wellknown that degree$d$ Chow parameters uniquely characterize degree$d$ polynomial threshold functions
(PTFs) within the space of all bounded functions. In this paper, we prove a robust version of this theorem:
For $f$ any Boolean degree$d$ PTF and $g$ any bounded function, if the degree$d$ Chow parameters of
$f$ are close to the degree$d$ Chow parameters of $g$ in $\ell_2$norm, then $f$ is close to $g$ in $\ell_1$distance.
Notably, our bound relating the two distances is independent of the dimension. That is,
we show that Boolean degree$d$ PTFs are {\em robustly identifiable} from their degree$d$ Chow parameters.
No nontrivial bound was previously known for $d >1$.
Our robust identifiability result gives the following algorithmic applications:
First, we show that Boolean degree$d$ PTFs can be efficiently approximately reconstructed
from approximations to their degree$d$ Chow parameters. This immediately implies
that degree$d$ PTFs are efficiently learnable in the uniform distribution $d$RFA model.
As a byproduct of our approach, we also obtain the first low integerweight approximations of degree$d$ PTFs, for $d>1$.
As our second application, our robust identifiability result gives the first efficient
algorithm, with dimensionindependent error guarantees,
for malicious learning of Boolean degree$d$ PTFs under the uniform distribution.
The proof of our robust identifiability result involves several new technical ingredients, including
the following structural result for degree$d$ multivariate polynomials with very poor anticoncentration:
If $p$ is a degree$d$ polynomial where $p(x)$ is {\em very} close to $0$ on a {\em large} number of points in $\bn$,
then there exists a degree$d$ hypersurface that exactly passes though {\em almost all} of these points.
We leverage this structural result to show that if the degree$d$ Chow distance between $f$ and $g$ is small,
then we can find many degree$d$ polynomials that vanish on their disagreement region,
and in particular enough that forces the $\ell_1$distance between $f$ and $g$ to also be small.
To implement this proof strategy, we require additional technical ideas.
In particular, in the $d=2$ case we show that for any large vector space of degree$2$ polynomials
with a large number of common zeroes, there exists a linear function that vanishes on almost all of these zeroes.
The degree$d$ degree generalization of this statement is significantly more complex, and can be viewed as an effective version
of Hilbert's Basis Theorem for our setting.

Efficient Robust Proper Learning of Logconcave Distributions
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Manuscript
We study the {\it robust proper learning} of univariate logconcave distributions
(over continuous and discrete domains). Given a set of samples drawn
from an unknown target distribution, we want to compute a logconcave
hypothesis distribution that is as close as possible to the target, in total variation distance.
In this work, we give the first computationally efficient algorithm
for this learning problem. Our algorithm achieves the informationtheoretically optimal
sample size (up to a constant factor), runs in polynomial time,
and is robust to model misspecification with nearlyoptimal
error guarantees.
Specifically, we give an algorithm that,
on input $n=O(1/\epsilon^{5/2})$ samples from an unknown distribution $f$,
runs in time $\widetilde{O}(n^{8/5})$,
and outputs a logconcave hypothesis $h$ that (with high probability) satisfies
$d_{\mathrm{TV}}(h, f) = O(\mathrm{opt})+\epsilon$, where $\mathrm{opt}$
is the minimum total variation distance between $f$
and the class of logconcave distributions.
Our approach to the robust proper learning problem is quite flexible and may be applicable
to many other univariate distribution families.

On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games
[abstract]
[arxiv]
I. Diakonikolas, C. Pavlou
Proceedings of the 33rd AAAI Conference on Artificial Intelligence (AAAI 2019)
Weighted voting games are a family of cooperative games,
typically used to model voting situations where a number of agents (players) vote against or for a proposal.
In such games, a proposal is accepted if an appropriately weighted sum of the votes
exceeds a prespecified threshold. As the influence of a player over the voting outcome
is not in general proportional to her assigned weight,
various power indices have been proposed to measure each player's influence.
The inverse power index problem is the problem of designing a weighted
voting game that achieves a set of target influences according to a predefined power index.
In this work, we study the computational complexity of the inverse problem when
the power index belongs to the class of semivalues. We prove that the inverse problem
is computationally intractable for a broad family of semivalues, including all regular semivalues.
As a special case of our general result, we establish computational hardness
of the inverse problem for the Banzhaf indices and the Shapley values,
arguably the most popular power indices.

Collisionbased Testers are Optimal for Uniformity and Closeness
[abstract]
[eccc]
I. Diakonikolas, T. Gouleakis, J. Peebles, E. Price
Chicago Journal of Theoretical Computer Science, 2019
Also see Oded Goldreich's exposition of our proof
here
We study the fundamental problems of (i) uniformity testing of a discrete distribution,
and (ii) closeness testing between two discrete distributions with bounded $\ell_2$norm.
These problems have been extensively studied in distribution testing
and sampleoptimal estimators are known for them~\cite{Paninski:08, CDVV14, VV14, DKN:15}.
In this work, we show that the original collisionbased testers proposed for these problems
~\cite{GRdist:00, BFR+:00} are sampleoptimal, up to constant factors.
Previous analyses showed sample complexity upper bounds for these testers that are optimal
as a function of the domain size $n$, but suboptimal by polynomial factors
in the error parameter $\epsilon$. Our main contribution is a new tight analysis
establishing that these collisionbased testers are informationtheoretically optimal,
up to constant factors, both in the dependence on $n$ and in the dependence on $\epsilon$.

HighDimensional Robust Mean Estimation in NearlyLinear Time
[abstract]
[arxiv]
Y. Cheng, I. Diakonikolas, R. Ge
Proceedings of the 30th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2019)
We study the fundamental problem of highdimensional mean estimation in a robust model
where a constant fraction of the samples are adversarially corrupted.
Recent work gave the first polynomial time algorithms for this problem
with dimensionindependent error guarantees for several families of structured distributions.
In this work, we give the first nearlylinear time algorithms for highdimensional robust mean estimation.
Specifically, we focus on distributions with (i) known covariance and subgaussian tails, and (ii) unknown bounded covariance.
Given $N$ samples on $\mathbb{R}^d$, an $\epsilon$fraction of which may be arbitrarily corrupted, our algorithms run in time
$\tilde{O}(Nd) / \mathrm{poly}(\epsilon)$ and approximate the true mean within the informationtheoretically
optimal error, up to constant factors.
Previous robust algorithms with comparable error guarantees have running times
$\tilde{\Omega}(N d^2)$, for $\epsilon = \Omega(1)$.
Our algorithms rely on a natural family of SDPs parameterized by our current guess $\nu$
for the unknown mean $\mu^\star$.
We give a winwin analysis establishing the following: either a nearoptimal
solution to the primal SDP yields a good candidate
for $\mu^\star$  independent of our current guess $\nu$  or the dual SDP yields
a new guess $\nu'$ whose distance from $\mu^\star$ is smaller by a constant factor.
We exploit the special structure of the corresponding SDPs to show that they
are approximately solvable in nearlylinear time.
Our approach is quite general, and we believe it can also be applied to obtain nearlylinear time algorithms
for other highdimensional robust learning problems.

Efficient Algorithms and Lower Bounds for Robust Linear Regression
[abstract]
[arxiv]
I. Diakonikolas, W. Kong, A. Stewart
Proceedings of the 30th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2019)
We study the prototypical problem of highdimensional linear regression in a robust model where an $\epsilon$fraction
of the samples can be adversarially corrupted. We focus on the fundamental setting where the covariates
of the uncorrupted samples are drawn from a Gaussian distribution $\mathcal{N}(0, \Sigma)$ on $\R^d$.
We give nearly tight upper bounds and computational lower bounds for this problem.
Specifically, our main contributions are as follows:
For the case that the covariance matrix is known to be the identity,
we give a sample nearoptimal and computationally efficient algorithm that draws $\tilde{O}(d/\eps^2)$
labeled examples and outputs a candidate hypothesis vector $\widehat{\beta}$ that approximates the unknown regression vector
$\beta$ within $\ell_2$norm $O(\eps \log(1/\eps) \sigma)$, where $\sigma$ is the standard deviation
of the random observation noise. An error of $\Omega (\eps \sigma)$ is informationtheoretically
necessary, even with infinite sample size. Hence, the error guarantee of our algorithm is
optimal, up to a logarithmic factor in $1/\eps$.
Prior work gave an algorithm for this problem with sample complexity $\tilde{\Omega}(d^2/\eps^2)$
whose error guarantee scales with the $\ell_2$norm of $\beta$.
For the case of unknown covariance $\Sigma$,
we show that we can efficiently achieve the same error guarantee of $O(\eps \log(1/\eps) \sigma)$,
as in the known covariance case, using an additional $\tilde{O}(d^2/\eps^2)$ unlabeled examples.
On the other hand, an error of $O(\eps \sigma)$ can be informationtheoretically attained with $O(d/\eps^2)$ samples.
We prove a Statistical Query (SQ) lower bound providing evidence that this quadratic
tradeoff in the sample size is inherent. More specifically, we show that any polynomial time
SQ learning algorithm for robust linear regression (in Huber's contamination model)
with estimation complexity $O(d^{2c})$, where $c>0$ is an arbitrarily small constant,
must incur an error of $\Omega(\sqrt{\eps} \sigma)$.

Fourierbased Testing for Families of Distributions
[abstract]
[eccc]
C. Canonne, I. Diakonikolas, A. Stewart
Advances in Neural Information Processing Systems (NeurIPS 2018)
We study the general problem of testing whether an unknown discrete distribution belongs to a given family of distributions.
More specifically, given a class of distributions $\mathcal{P}$ and sample access to an unknown distribution $p$,
we want to distinguish (with high probability) between the case that $p \in \mathcal{P}$ and the case
that $p$ is $\epsilon$far, in total variation distance, from every distribution in $\mathcal{P}$.
This is the prototypical hypothesis testing problem that has received significant attention in statistics and,
more recently, in theoretical computer science.
The sample complexity of this general problem depends on the underlying family $\mathcal{P}$.
We are interested in designing sampleoptimal and computationally efficient algorithms for this task.
The main contribution of this work is a new and simple testing technique that is applicable to distribution families
whose \emph{Fourier spectrum} approximately satisfies a certain \emph{sparsity} property. As the main applications
of our Fourierbased testing technique, we obtain the first nontrivial testers for two fundamental families of discrete distributions:
Sums of Independent Integer Random Variables (SIIRVs) and Poisson Multinomial Distributions (PMDs).
Our testers for these families are nearly sampleoptimal and computationally efficient. We also obtain
a tester with improved sample complexity for discrete logconcave distributions.
To the best of our knowledge, ours is the first use of the Fourier transform in the context of distribution testing.

Sharp Bounds for Generalized Uniformity Testing
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Advances in Neural Information Processing Systems (NeurIPS 2018)
Selected for Spotlight Presentation at NeurIPS 2018
See the comments in Oded Goldreich's Choices
#229
We study the problem of {\em generalized uniformity testing}~\cite{BC17} of a discrete probability distribution:
Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbf{\Omega}$,
we want to distinguish, with probability at least $2/3$, between the case that $p$
is uniform on some {\em subset} of $\mathbf{\Omega}$ versus $\epsilon$far,
in total variation distance, from any such uniform distribution.
We establish tight bounds on the sample complexity of generalized uniformity testing.
In more detail, we present a computationally efficient tester
whose sample complexity is optimal, up to constant factors,
and a matching informationtheoretic lower bound.
Specifically, we show that the sample complexity of generalized uniformity testing is
$\Theta\left(1/(\epsilon^{4/3}\p\_3) + 1/(\epsilon^{2} \p\_2) \right)$.

Robust Learning of FixedStructure Bayesian Networks
[abstract]
[arxiv]
Y. Cheng, I. Diakonikolas, D. Kane, A. Stewart
Advances in Neural Information Processing Systems (NeurIPS 2018)
We investigate the problem of learning Bayesian networks in an agnostic model
where an $\epsilon$fraction of the samples are adversarially corrupted.
Our agnostic learning model is similar to  in fact, stronger than  Huber's
contamination model in robust statistics. In this work, we study the fully observable
Bernoulli case where the structure of the network is given.
Even in this basic setting, previous learning algorithms
either run in exponential time or lose dimensiondependent factors in their
error guarantees.
We provide the first computationally efficient agnostic learning algorithm for this problem
with dimensionindependent error guarantees. Our algorithm has polynomial sample complexity,
runs in polynomial time, and achieves error that scales nearlylinearly with the fraction
of adversarially corrupted samples.

Differentially Private Identity and Closeness Testing of Discrete Distributions
[abstract]
[arxiv]
M. Aliakbarpour, I. Diakonikolas, R. Rubinfeld
Proceedings of the 35th International Conference on Machine Learning (ICML 2018)
We investigate the problems of identity and closeness testing over a discrete population
from random samples. Our goal is to develop efficient testers while guaranteeing
Differential Privacy to the individuals of the population. We describe an approach that
yields sampleefficient differentially private testers for these problems.
Our theoretical results show that there exist private identity and closeness testers
that are nearly as sampleefficient as their nonprivate counterparts. We perform
an experimental evaluation of our algorithms on synthetic data. Our experiments
illustrate that our private testers achieve small type I and type II errors with sample size
{\em sublinear} in the domain size of the underlying distributions.

NearOptimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Logconcave Densities
[abstract]
[arxiv]
T. Carpenter, I. Diakonikolas, A. Sidiropoulos, A. Stewart
Proceedings of the 31st Annual Conference on Learning Theory (COLT 2018)
We study the problem of learning multivariate logconcave densities
with respect to a global loss function. We obtain the first upper bound on the sample complexity
of the maximum likelihood estimator (MLE) for a logconcave density on $\mathbb{R}^d$, for all $d \geq 4$.
Prior to this work, no finite sample upper bound was known for this estimator in more than $3$ dimensions.
In more detail, we prove that for any $d \geq 1$ and $\epsilon>0$, given
$\tilde{O}_d((1/\epsilon)^{(d+3)/2})$ samples drawn from an unknown logconcave density $f_0$ on $\mathbb{R}^d$,
the MLE outputs a hypothesis $h$ that with high probability is $\epsilon$close
to $f_0$, in squared Hellinger loss. A sample complexity lower bound of $\Omega_d((1/\epsiilon)^{(d+1)/2})$
was previously known for any learning algorithm that achieves this guarantee.
We thus establish that the sample complexity of the logconcave MLE is nearoptimal,
up to an $\tilde{O}(1/\epsilon)$ factor.

Fast and Sample NearOptimal Algorithms for Learning Multidimensional Histograms
[abstract]
[arxiv]
I. Diakonikolas, J. Li, L. Schmidt
Proceedings of the 31st Annual Conference on Learning Theory (COLT 2018)
We study the problem of robustly learning multidimensional histograms.
A $d$dimensional function $h: D \to \R$ is called a $k$histogram if there exists a partition of the
domain $D \subseteq \R^d$ into $k$ axisaligned rectangles such that $h$ is constant within each such rectangle.
Let $f: D \to \R$ be a $d$dimensional probability density function
and suppose that $f$ is $\mathrm{OPT}$close, in $L_1$distance,
to an unknown $k$histogram (with unknown partition). Our goal is to output a hypothesis
that is $O(\mathrm{OPT}) + \epsilon$ close to $f$, in $L_1$distance. We give an algorithm for this learning
problem that uses $n = \tilde{O}_d(k/\eps^2)$ samples and runs in time $\tilde{O}_d(n)$.
For any fixed dimension, our algorithm has optimal sample complexity, up to logarithmic factors,
and runs in nearlinear time. Prior to our work, the time complexity of the $d=1$ case was wellunderstood,
but significant gaps in our understanding remained even for $d=2$.

SampleOptimal Identity Testing with High Probability
[abstract]
[arxiv]
I. Diakonikolas, T. Gouleakis, J. Peebles, E. Price
Proceedings of the 45th Intl. Colloquium on Automata, Languages and Programming (ICALP 2018)
See the comments in Oded Goldreich's Choices
#229
We study the problem of testing identity against a given
distribution (a.k.a. goodnessoffit) with a focus on the high
confidence regime. More precisely, given samples from
an unknown distribution $p$ over $n$ elements, an explicitly given
distribution $q$, and parameters $0< \epsilon, \delta < 1$, we wish
to distinguish, {\em with probability at least $1\delta$}, whether
the distributions are identical versus $\epsilon$far in total variation (or statistical) distance.
Existing work has focused on
the constant confidence regime, i.e., the case that
$\delta = \Omega(1)$, for which the sample complexity of
identity testing is known to be $\Theta(\sqrt{n}/\epsilon^2)$.
Typical applications of distribution property testing
require small values of the confidence parameter $\delta$ (which correspond to small
``$p$values'' in the statistical hypothesis testing terminology). Prior work achieved arbitrarily small values
of $\delta$ via blackbox amplification, which multiplies the required number of samples by
$\Theta(\log(1/\delta))$. We show that this upper bound is suboptimal for any
$\delta = o(1)$, and give a new identity tester that achieves the
optimal sample complexity. Our new upper and lower bounds show that
the optimal sample complexity of identity testing is
\[
\Theta\left( \frac{1}{\epsilon^2}\left(\sqrt{n \log(1/\delta)} + \log(1/\delta) \right)\right)
\]
for any $n, \epsilon$, and $\delta$. For the special case of uniformity
testing, where the given distribution is the uniform distribution $U_n$ over the domain,
our new tester is surprisingly simple: to test whether $p = U_n$ versus $\mathrm{d}_{TV}(p, U_n) \geq \epsilon$,
we simply threshold $\mathrm{d}_{TV}(\hat{p}, U_n)$, where $\hat{p}$ is the empirical probability distribution.
We believe that our novel analysis techniques may be useful for
other distribution testing problems as well.

Testing Conditional Independence of Discrete Distributions
[abstract]
[arxiv]
C. Canonne, I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC 2018)
We study the problem of testing \emph{conditional independence} for discrete distributions.
Specifically, given samples from a discrete random variable $(X, Y, Z)$ on domain $[\ell_1]\times[\ell_2] \times [n]$,
we want to distinguish, with probability at least $2/3$, between the case that $X$ and $Y$ are conditionally independent
given $Z$ from the case that $(X, Y, Z)$ is $\eps$far, in $\lp[1]$distance, from every distribution that has this property.
Conditional independence is a concept of central importance in probability and statistics with a range of applications
in various scientific domains. As such, the statistical task of testing conditional independence has been extensively studied
in various forms within the statistics and econometrics communities for nearly a century.
Perhaps surprisingly, this problem has not been previously considered in the framework of distribution property testing
and in particular no tester with sublinear sample complexity is known, even for the important special case that the domains of $X$ and $Y$ are binary.
The main algorithmic result of this work is the first conditional independence tester with {\em sublinear} sample complexity for
discrete distributions over $[\ell_1]\times[\ell_2] \times [n]$.
To complement our upper bounds, we prove informationtheoretic lower bounds establishing
that the sample complexity of our algorithm is optimal, up to constant factors, for a number of settings.
Specifically, for the prototypical setting when $\ell_1, \ell_2 = O(1)$, we show that the sample complexity of testing
conditional independence (upper bound and matching lower bound) is
\[
\bigTheta{\max\left(n^{1/2}/\eps^2,\min\mleft(n^{7/8}/\eps,n^{6/7}/\eps^{8/7}\mright)\right)}\,.
\]
To obtain our tester, we employ a variety of tools, including
(1) a suitable weighted adaptation of the flattening technique~\cite{DK:16},
and (2) the design and analysis of an optimal (unbiased) estimator
for the following statistical problem of independent interest:
Given a degree$d$ polynomial $Q\colon\mathbb{R}^n \to \R$
and sample access to a distribution $p$ over $[n]$,
estimate $Q(p_1, \ldots, p_n)$ up to small additive error.
Obtaining tight variance analyses for specific estimators of this form
has been a major technical hurdle in distribution testing (see, e.g.,~\cite{CDVV14}).
As an important contribution of this work, we develop a general theory
providing tight variance bounds for \emph{all} such estimators. Our lower bounds, established
using the mutual information method, rely on novel constructions of hard instances
that may be useful in other settings.

ListDecodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussians
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC 2018)
We study the problem of {\emph listdecodable (robust) Gaussian mean estimation} and the related problem
of {\emph learning mixtures of separated spherical Gaussians}. In the former problem, we are given a set $T$
of points in $\mathbb{R}^n$ with the promise that an $\alpha$fraction of points in $T$, where $0< \alpha < 1/2$,
are drawn from an unknown mean identity covariance Gaussian $G$,
and no assumptions are made about the remaining points.
The goal is to output a small list of candidate vectors with the guarantee that at least one of
the candidates is close to the mean of $G$. In the latter problem, we are given samples from a $k$mixture
of spherical Gaussians on $\mathbb{R}^n$ and the goal is to estimate the unknown model parameters up to small accuracy.
We develop a set of techniques that yield new efficient algorithms with significantly improved
guarantees for these problems. Specifically, our main contributions are as follows:
ListDecodable Mean Estimation. Fix any $d \in \mathbb{Z}_+$ and $0< \alpha <1/2$.
We design an algorithm with sample complexity $O_d (\mathrm{poly}(n^d/\alpha))$ and runtime
$O_d (\mathrm{poly}(n/\alpha)^{d})$
that outputs a list of $O(1/\alpha)$ many candidate vectors such that with high probability
one of the candidates is within $\ell_2$distance $O_d(\alpha^{1/(2d)})$ from the mean of $G$.
The only previous algorithm for this problem~\cite{CSV17}
achieved error $\tilde O(\alpha^{1/2})$ under second moment conditions.
For $d = O(1/\eps)$, where $\eps>0$ is a constant,
our algorithm runs in polynomial time and achieves error $O(\alpha^{\eps})$.
For $d = \Theta(\log(1/\alpha))$, our algorithm runs in time $(n/\alpha)^{O(\log(1/\alpha))}$
and achieves error $O(\log^{3/2}(1/\alpha))$, almost matching the informationtheoretically
optimal bound of $\Theta(\log^{1/2}(1/\alpha))$ that we establish.
We also give a Statistical Query (SQ) lower bound
suggesting that the complexity of our algorithm is qualitatively close to best possible.
Learning Mixtures of Spherical Gaussians. We give a learning algorithm
for mixtures of spherical Gaussians,
with unknown spherical covariances, that succeeds under significantly weaker
separation assumptions compared to prior work. For the prototypical case
of a uniform $k$mixture of identity covariance Gaussians we obtain the following:
For any $\eps>0$, if the pairwise separation between the means is at least
$\Omega(k^{\epsilon}+\sqrt{\log(1/\delta)})$, our algorithm learns the unknown parameters
within accuracy $\delta$ with sample complexity and running time $\poly (n, 1/\delta, (k/\epsilon)^{1/\epsilon})$.
Moreover, our algorithm is robust to a small dimensionindependent
fraction of corrupted data. The previously best known polynomial time algorithm~\cite{VempalaWang:02}
required separation at least $k^{1/4} \polylog(k/\delta)$.
Finally, our algorithm works under separation of $\new{\tilde O(\log^{3/2}(k)+\sqrt{\log(1/\delta)})}$
with sample complexity and running time $\poly(n, 1/\delta, k^{\log k})$.
This bound is close to the informationtheoretically minimum separation of $\Omega(\sqrt{\log k})$~\cite{RV17}.
Our main technical contribution is a new technique, using degree$d$ multivariate polynomials,
to remove outliers from highdimensional datasets where the majority of the points are corrupted.

Learning Geometric Concepts with Nasty Noise
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC 2018)
We study the efficient learnability of geometric concept classes 
specifically, lowdegree polynomial threshold functions (PTFs)
and intersections of halfspaces  when a fraction of the
training data is adversarially corrupted. We give the first polynomialtime
PAC learning algorithms for these concept classes with {\em dimensionindependent} error guarantees
in the presence of {\em nasty noise} under the Gaussian distribution. In the nasty noise model,
an omniscient adversary can arbitrarily corrupt a small fraction of both the unlabeled data points and their labels.
This model generalizes wellstudied noise models,
including the malicious noise model and the agnostic (adversarial label noise) model.
Prior to our work, the only concept class for which efficient malicious learning
algorithms were known was the class of {\em origincentered} halfspaces.
Specifically, our robust learning algorithm for lowdegree PTFs
succeeds under a number of tame distributions  including the Gaussian distribution
and, more generally, any logconcave distribution with (approximately) known lowdegree moments.
For LTFs under the Gaussian distribution, we give
a polynomialtime algorithm that achieves error $O(\epsilon)$, where $\epsilon$ is the noise rate.
At the core of our PAC learning results is an efficient algorithm
to approximate the {\em lowdegree Chowparameters}
of any bounded function in the presence of nasty noise.
To achieve this, we employ an iterative spectral method for outlier detection and removal,
inspired by recent work in robust unsupervised learning.
Our aforementioned algorithm succeeds for a range of distributions satisfying
mild concentration bounds and moment assumptions.
The correctness of our robust learning algorithm for intersections of halfspaces
makes essential use of a novel robust inverse independence lemma
that may be of broader interest.

Robustly Learning a Gaussian: Getting Optimal Error, Efficiently
[abstract]
[arxiv]
I. Diakonikolas, G. Kamath, D. Kane, J. Li, A. Moitra, A. Stewart
Proceedings of the 29th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2018)
We study the fundamental problem of learning the parameters of a highdimensional Gaussian
in the presence of noise  where an $\epsilon$fraction of our samples were chosen by an adversary.
We give robust estimators that achieve estimation error $O(\epsilon)$ in the total variation distance,
which is optimal up to a universal constant that is independent of the dimension.
In the case where just the mean is unknown, our robustness guarantee is optimal up to a factor of $\sqrt{2}$
and the running time is polynomial in $d$ and $1/\epsilon$. When both the mean and covariance are unknown,
the running time is polynomial in $d$ and quasipolynomial in $1/\epsilon$. Moreover all of our algorithms
require only a polynomial number of samples. Our work shows that the same sorts of error guarantees
that were established over fifty years ago in the onedimensional setting can also be achieved
by efficient algorithms in highdimensional settings.

CommunicationEfficient Distributed Learning of Discrete Distributions
[abstract]
[proceedings]
I. Diakonikolas, E. Grigorescu, J. Li, A. Natarajan, K. Onak, L. Schmidt
Advances in Neural Information Processing Systems (NIPS 2017)
Selected for Oral Presentation at NIPS 2017
We initiate a systematic investigation of density estimation
when the data is {\em distributed} across multiple servers.
The servers must communicate with a referee and the goal is to estimate
the underlying distribution with as few bits of communication as possible.
We focus on nonparametric density estimation of discrete distributions
with respect to the $\ell_1$ and $\ell_2$ norms. We provide the first nontrivial
upper and lower bounds on the communication complexity of this basic estimation
task in various settings of interest.
When the unknown discrete distribution is {\em unstructured} and each server
has only one sample, we show that any {\em blackboard} protocol
(i.e., any protocol in which servers interact arbitrarily using public messages)
that learns the distribution must essentially communicate the entire sample.
For the case of {\em structured} distributions, such as $k$histograms and monotone distributions,
we design distributed learning algorithms that achieve significantly better communication
guarantees than the naive ones, and obtain tight upper and lower bounds in several regimes.
Our distributed learning algorithms run in nearlinear time and are robust to model misspecification.
Our results provide insights on the interplay between structure and communication efficiency for a range
of fundamental distribution estimation tasks.

Statistical Query Lower Bounds for Robust Estimation of Highdimensional Gaussians and Gaussian Mixtures
[abstract]
[eccc]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2017)
See here
and here
for videos of related talks.
We prove the first {\em Statistical Query lower bounds} for
two fundamental highdimensional learning problems involving
Gaussian distributions: (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning
of a single unknown mean Gaussian. In particular, we show a {\em superpolynomial gap} between the (informationtheoretic)
sample complexity and the complexity of {\em any} Statistical Query algorithm for these problems.
Statistical Query (SQ) algorithms are a class of algorithms
that are only allowed to query expectations of functions of the distribution rather than directly access samples.
This class of algorithms is quite broad: with the sole exception of Gaussian elimination over finite fields,
all known algorithmic approaches in machine learning can be implemented in this model.
Our SQ lower bound for Problem (1)
is qualitatively matched by known learning algorithms for GMMs (all of which can be implemented as SQ algorithms).
At a conceptual level, this result implies that  as far as SQ algorithms are concerned  the computational complexity
of learning GMMs is inherently exponential
{\it in the dimension of the latent space}  even though there
is no such informationtheoretic barrier. Our lower bound for Problem (2) implies that the accuracy of the robust learning algorithm
in~\cite{DiakonikolasKKLMS16} is essentially best possible among all polynomialtime SQ algorithms.
On the positive side, we give a new SQ learning algorithm for this problem
with optimal accuracy whose running time nearly matches our lower bound.
Both our SQ lower bounds are attained via a unified momentmatching technique that may be useful in other contexts.
Our SQ learning algorithm for Problem (2) relies on a filtering technique that removes outliers based on higherorder tensors.
Our lower bound technique also has implications for related inference problems,
specifically for the problem of robust {\it testing} of an unknown mean Gaussian.
Here we show an informationtheoretic lower bound
which separates the sample complexity of the robust testing problem from its nonrobust variant.
This result is surprising because such a separation does not exist
for the corresponding learning problem.

Being Robust (in High Dimensions) Can be Practical
[abstract]
[arxiv]
[code]
I. Diakonikolas, G. Kamath, D. Kane, J. Li, A. Moitra, A. Stewart
Proceedings of the 34th International Conference on Machine Learning (ICML 2017)
Robust estimation is much more challenging in high dimensions than it is in one dimension:
Most techniques either lead to intractable optimization problems or estimators
that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that,
in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time
algorithms that can tolerate a constant fraction of corruptions, independent of the dimension.
However, the sample and time complexity of these algorithms is prohibitively large for highdimensional applications.
In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors,
as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions.
Finally, we show on both synthetic and real data that our algorithms have stateoftheart performance and suddenly make
highdimensional robust estimation a realistic possibility.

Testing Bayesian Networks
[abstract]
[arxiv]
C. Canonne, I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 30th Annual Conference on Learning Theory (COLT 2017)
Journal version in IEEE Transactions on Information Theory, to appear.
This work initiates a systematic investigation of testing {\em highdimensional} structured
distributions by focusing on testing {\em Bayesian networks} 
the prototypical family of directed graphical models. A Bayesian network
is defined by a directed acyclic graph, where we associate a random variable with each node.
The value at any particular node is conditionally independent of all the other nondescendant nodes once its parents are fixed.
Specifically, we study the properties of identity testing and closeness testing of Bayesian networks. Our main contribution is
the first nontrivial efficient testing algorithms for these problems and corresponding informationtheoretic lower bounds.
For a wide range of parameter settings, our testing algorithms have sample complexity {\em sublinear} in the dimension
and are sampleoptimal, up to constant factors.

Learning Multivariate Logconcave Distributions
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 30th Annual Conference on Learning Theory (COLT 2017)
We study the problem of estimating multivariate logconcave probability density functions.
We prove the first sample complexity upper bound for learning logconcave densities
on $\mathbb{R}^d$, for all $d \geq 1$. Prior to our work, no upper bound on the
sample complexity of this learning problem was known for the case of $d>3$.
In more detail, we give an estimator that, for any $d \ge 1$ and $\epsilon>0$,
draws $\tilde{O}_d \left( (1/\epsilon)^{(d+5)/2} \right)$ samples from an unknown
target logconcave density on $\mathbb{R}^d$, and outputs a hypothesis that
(with high probability) is $\epsilon$close to the target, in total variation distance.
Our upper bound on the sample complexity comes close to the known lower bound of
$\Omega_d \left( (1/\epsilon)^{(d+1)/2} \right)$ for this problem.

Nearoptimal Closeness Testing of Discrete Histogram Distributions
[abstract]
[arxiv]
I. Diakonikolas, D. Kane, V. Nikishkin
Proceedings of the 44th Intl. Colloquium on Automata, Languages and Programming (ICALP 2017)
We investigate the problem of testing the equivalence between two discrete histograms.
A {\em $k$histogram} over $[n]$ is a probability distribution that is piecewise constant over some set of $k$ intervals over $[n]$.
Histograms have been extensively studied in computer science and statistics.
Given a set of samples from two $k$histogram distributions $p, q$ over $[n]$,
we want to distinguish (with high probability) between the cases that $p = q$ and $\pq\_1 \geq \epsilon$.
The main contribution of this paper is a new algorithm for this testing problem
and a nearly matching informationtheoretic lower bound.
Specifically, the sample complexity of our algorithm matches our lower bound up to a logarithmic factor, improving
on previous work by polynomial factors in the relevant parameters.
Our algorithmic approach applies in a more general framework and yields improved sample upper bounds
for testing closeness of other structured distributions as well.

Playing Anonymous Games using Simple Strategies
[abstract]
[arxiv]
Y. Cheng, I. Diakonikolas, A. Stewart
Proceedings of the 28th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2017)
We investigate the complexity of computing approximate Nash equilibria in anonymous games.
Our main algorithmic result is the following: For any $n$player anonymous game with a bounded
number of strategies and any constant $\delta>0$, an $O(1/n^{1\delta})$approximate Nash
equilibrium can be computed in polynomial time.
Complementing this positive result, we show that if there exists any constant $\delta>0$
such that an $O(1/n^{1+\delta})$approximate equilibrium can be computed in polynomial time,
then there is a fully polynomialtime approximation scheme for this problem.
We also present a faster algorithm that, for any $n$player $k$strategy anonymous game,
runs in time $\tilde O((n+k) k n^k)$ and computes an $\tilde O(n^{1/3} k^{11/3})$approximate equilibrium.
This algorithm follows from the existence of simple approximate equilibria of anonymous games,
where each player plays one strategy with probability $1\delta$, for some small $\delta$,
and plays uniformly at random with probability $\delta$.
Our approach exploits the connection between Nash equilibria in anonymous games
and Poisson multinomial distributions (PMDs).
Specifically, we prove a new probabilistic lemma establishing the following:
Two PMDs, with large variance in each direction, whose first few moments
are approximately matching are close in total variation distance.
Our structural result strengthens previous work by providing a smooth tradeoff
between the variance bound and the number of matching moments.

Sample Optimal Density Estimation in NearlyLinear Time
[abstract]
[pdf]
[code]
J. Acharya, I. Diakonikolas, J. Li, L. Schmidt
Proceedings of the 28th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2017)
We design a new, fast algorithm for agnostically learning univariate probability distributions
whose densities are well approximated by piecewise polynomial functions.
Let $f$ be the density function of an arbitrary univariate distribution,
and suppose that $f$ is $\mathrm{OPT}$ close in $L_1$ distance to an unknown piecewise polynomial function
with $t$ interval pieces and degree $d$. Our algorithm
draws $m = O(t(d+1)/\epsilon^2)$ samples from $f$, runs in time
$\widetilde{O} (m \cdot \mathrm{poly} (d))$ and with probability at least
$9/10$ outputs an $O(t)$piecewise degree$m$ hypothesis $h$ that is
$4 \mathrm{OPT} +\epsilon$ close to $f$.
Our general algorithm yields (near)sampleoptimal and nearlinear time estimators
for a wide range of structured distribution families
over both continuous and discrete domains in a unified way. For most of our applications,
these are the first sampleoptimal and nearlinear time
estimators in the literature. As a consequence, our work resolves the sample
and computational complexities of a broad class of inference
tasks via a single "metaalgorithm". Moreover, we experimentally demonstrate
that our algorithm performs very well in practice.

Robust Estimators in High Dimensions without the Computational Intractability
[abstract]
[arxiv]
I. Diakonikolas, G. Kamath, D. Kane, J. Li, A. Moitra, A. Stewart
Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016)
SIAM Journal on Computing Special Issue for FOCS 2016
Invited to Communications of the ACM, Research Highlights
See here
for my Simons Institute tutorial on the topic and here for a previous related talk.
We study highdimensional distribution learning in an agnostic setting
where an adversary is allowed to arbitrarily corrupt an $\epsilon$ fraction of the samples.
Such questions have a rich history spanning statistics, machine learning and theoretical computer science.
Even in the most basic settings,
the only known approaches are either computationally inefficient
or lose dimension dependent factors in their error guarantees.
This raises the following question:
Is highdimensional agnostic distribution learning even possible, algorithmically?
In this work, we obtain the first computationally efficient algorithms with dimensionindependent error guarantees
for agnostically learning several fundamental classes of highdimensional distributions:
(1) a single Gaussian, (2) a product distribution on the hypercube,
(3) mixtures of two product distributions (under a natural balancedness condition),
and (4) mixtures of spherical Gaussians.
Our algorithms achieve error that is independent of the dimension,
and in many cases scales nearlylinearly
with the fraction of adversarially corrupted samples.
Moreover, we develop a general recipe for detecting and correcting corruptions in high dimensions,
that may be applicable to many other problems.

A New Approach for Testing Properties of Discrete Distributions
[abstract]
[pdf]
[arxiv]
I. Diakonikolas, D. Kane
Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016)
Also see the comments in Oded Goldreich's Choices
#188 and
#195,
and Oded's very nice exposition of our framework here
We study problems in distribution property testing:
Given sample access to one or more unknown discrete distributions,
we want to determine whether they have some global property or are $\epsilon$far
from having the property in $\ell_1$ distance.
In this paper, we provide a simple and general approach to obtain upper bounds in this setting,
by reducing $\ell_1$testing to $\ell_2$testing.
Our reduction yields optimal $\ell_1$testers, by using a standard $\ell_2$tester as a blackbox.
Using our framework, we obtain sampleoptimal and computationally efficient estimators for
a wide variety of $\ell_1$ distribution testing problems, including the following:
identity testing to a fixed distribution,
closeness testing between two unknown distributions (with equal/unequal sample sizes),
independence testing (in any number of dimensions),
closeness testing for collections of distributions, and testing $k$histograms.
For most of these problems, we give the first optimal testers in the literature.
Moreover, our estimators are significantly simpler to state
and analyze compared to previous approaches.
As our second main contribution, we provide a direct general approach
for proving distribution testing lower bounds,
by bounding the mutual information. Our lower bound approach
is not restricted to symmetric properties,
and we use it to prove tight lower bounds for the aforementioned problems.

Fast Algorithms for Segmented Regression
[abstract]
[pdf]
[code]
J. Acharya, I. Diakonikolas, J. Li, L. Schmidt
Proceedings of the 33rd International Conference on Machine Learning (ICML 2016)
We study the fixed design segmented regression problem:
Given noisy samples from a piecewise linear function $f$,
we want to recover $f$ up to a desired accuracy in meansquared error.
Previous rigorous approaches for this problem rely on dynamic programming (DP)
and, while sample efficient, have running time quadratic in the sample size.
As our main contribution, we provide new
sample nearlinear time algorithms for the problem that 
while not being minimax optimal 
achieve a significantly better sampletime tradeoff
on large datasets compared to the DP approach.
Our experimental evaluation shows that, compared with the DP approach,
our algorithms provide a convergence rate that is only off by a factor of $2$ to $3$,
while achieving speedups of two orders of magnitude.

Properly Learning Poisson Binomial Distributions in Almost Polynomial Time
[abstract]
[pdf]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 29th Annual Conference on Learning Theory (COLT 2016)
We give an algorithm for properly learning Poisson binomial distributions.
A Poisson binomial distribution (PBD) of order $n$
is the discrete probability distribution of the sum of $n$ mutually independent Bernoulli random variables.
Given $\widetilde{O}(1/\epsilon^2)$ samples from an unknown PBD $\mathbf{P}$, our algorithm runs in time
$(1/\epsilon)^{O(\log \log (1/\epsilon))}$, and outputs a hypothesis PBD
that is $\epsilon$close to $\mathbf{P}$ in total variation distance.
The sample complexity of our algorithm is known to be nearlyoptimal,
up to logarithmic factors, as established
in previous work~\cite{DDS12stoc}. However, the previously best known running time for properly
learning PBDs~\cite{DDS12stoc, DKS15} was $(1/\epsilon)^{O(\log(1/\epsilon))}$,
and was essentially obtained by enumeration over an appropriate $\epsilon$cover.
We remark that the running time of this coverbased approach cannot be
improved, as any $\epsilon$cover for the space of PBDs
has size $(1/\epsilon)^{\Omega(\log(1/\epsilon))}$~\cite{DKS15}.
As one of our main contributions, we provide a novel structural characterization of PBDs,
showing that any PBD $\mathbf{P}$ is $\epsilon$close to another PBD $\mathbf{Q}$
with $O(\log(1/\epsilon))$ distinct parameters.
More precisely, we prove that, for all $\epsilon >0,$ there exists
an explicit collection $\cal{M}$ of $(1/\epsilon)^{O(\log \log (1/\epsilon))}$ vectors of multiplicities,
such that for any PBD $\mathbf{P}$ there exists a PBD $\mathbf{Q}$ with $O(\log(1/\epsilon))$
distinct parameters whose multiplicities are given by some element of ${\cal M}$,
such that $\mathbf{Q}$ is $\epsilon$close to $\mathbf{P}.$
Our proof combines tools from Fourier analysis and algebraic geometry.
Our approach to the proper learning problem is as follows:
Starting with an accurate nonproper hypothesis, we fit a PBD to this hypothesis.
More specifically, we essentially start with the hypothesis computed by the
computationally efficient nonproper learning algorithm in our recent work~\cite{DKS15}.
Our aforementioned structural characterization allows
us to reduce the corresponding fitting problem
to a collection of $(1/\epsilon)^{O(\log \log(1/\epsilon))}$
systems of lowdegree polynomial inequalities.
We show that each such system can be solved in time $(1/\epsilon)^{O(\log \log(1/\epsilon))}$,
which yields the overall running time of our algorithm.

Optimal Learning via the Fourier Transform for Sums of Independent Integer Random Variables
[abstract]
[pdf]
[arxiv]
[notes]
I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 29th Annual Conference on Learning Theory (COLT 2016)
We study the structure and learnability of sums of independent integer random variables (SIIRVs).
For $k \in \mathbb{Z}_{+}$, a {\em $k$SIIRV of order $n \in \mathbb{Z}_{+}$}
is the probability distribution of the sum of $n$
mutually independent random variables each supported on $\{0, 1, \dots, k1\}$.
We denote by ${\cal S}_{n,k}$ the set of all $k$SIIRVs of order $n$.
How many samples are required to learn an arbitrary distribution in ${\cal S}_{n,k}$?
In this paper, we tightly characterize the sample and computational complexity of this problem.
More precisely, we design a computationally efficient algorithm that uses $\widetilde{O}(k/\epsilon^2)$ samples,
and learns an arbitrary $k$SIIRV within error $\epsilon,$ in total variation distance. Moreover, we show that
the {\em optimal} sample complexity of this learning problem is
$\Theta((k/\epsilon^2)\sqrt{\log(1/\epsilon)}),$ i.e., we prove an upper bound and a matching
informationtheoretic lower bound.
Our algorithm proceeds by learning the Fourier transform of the target $k$SIIRV in its effective support.
Its correctness relies on the {\em approximate sparsity} of the Fourier transform of $k$SIIRVs 
a structural property that we establish, roughly stating that the Fourier transform of $k$SIIRVs
has small magnitude outside a small set.
Along the way we prove several new structural results about $k$SIIRVs.
As one of our main structural contributions, we give an efficient algorithm to construct a
sparse {\em proper} $\epsilon$cover for ${\cal S}_{n,k},$ in total variation distance.
We also obtain a novel geometric characterization of the space of $k$SIIRVs. Our
characterization allows us to prove a tight lower bound on the size of $\epsilon$covers for ${\cal S}_{n,k}$
 establishing that our cover upper bound is optimal  and is the key ingredient in our tight sample complexity lower bound.
Our approach of exploiting the sparsity of the Fourier transform in
distribution learning is general, and has recently found additional applications.
In a subsequent work~\cite{DKS15c}, we use a generalization of this idea (in higher dimensions)
to obtain the first efficient learning algorithm for Poisson multinomial distributions.
In~\cite{DKS15b}, we build on this approach to obtain the fastest known proper learning algorithm
for Poisson binomial distributions ($2$SIIRVs).

The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications
[abstract]
[pdf]
[arxiv]
I. Diakonikolas, D. Kane, A. Stewart
Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)
We study Poisson Multinomial Distributions  a fundamental family of discrete distributions
that generalize the binomial and multinomial distributions, and are commonly encountered
in computer science.
Formally, an $(n, k)$Poisson Multinomial Distribution (PMD) is a random variable
of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported
on the set $\{e_1, e_2, \ldots, e_k \}$ of standard basis vectors in $\mathbb{R}^k$.
In this paper, we obtain a refined structural understanding of PMDs
by analyzing their Fourier transform.
As our core structural result, we prove that the Fourier transform of PMDs is \emph{approximately sparse},
i.e., roughly speaking, its $L_1$norm is small outside a small set.
By building on this result, we obtain the following applications:
Learning Theory.
We design the first computationally efficient learning algorithm for PMDs
with respect to the total variation distance. Our algorithm learns an arbitrary $(n, k)$PMD
within variation distance $\epsilon$ using a nearoptimal sample size of $\widetilde{O}_k(1/\epsilon^2),$
and runs in time $\widetilde{O}_k(1/\epsilon^2) \cdot \log n.$ Previously, no algorithm with a $\mathrm{poly}(1/\epsilon)$
runtime was known, even for $k=3.$
Game Theory. We give the first efficient polynomialtime approximation scheme (EPTAS) for computing Nash equilibria
in anonymous games. For normalized anonymous games
with $n$ players and $k$ strategies, our algorithm computes a wellsupported $\epsilon$Nash equilibrium in time
$n^{O(k^3)} \cdot (k/\epsilon)^{O(k^3\log(k/\epsilon)/\log\log(k/\epsilon))^{k1}}.$
The best previous algorithm for this problem~\cite{DaskalakisP08, DaskalakisP2014}
had running time $n^{(f(k)/\epsilon)^k},$ where $f(k) = \Omega(k^{k^2})$, for any $k>2.$
Statistics. We prove a multivariate central limit theorem (CLT) that relates
an arbitrary PMD to a discretized multivariate Gaussian with the same mean and covariance, in total variation distance.
Our new CLT strengthens the CLT of Valiant and Valiant~\cite{VV10b, ValiantValiant:11} by completely removing the dependence on $n$ in the error bound.
Along the way we prove several new structural results of independent interest about PMDs.
These include: (i) a robust momentmatching lemma,
roughly stating that two PMDs that approximately agree on their lowdegree parameter moments are close in variation distance;
(ii) nearoptimal size proper $\epsilon$covers for PMDs in total variation distance
(constructive upper bound and nearlymatching lower bound).
In addition to Fourier analysis, we employ a number of analytic tools,
including the saddlepoint method from complex analysis,
that may find other applications.

Testing Shape Restrictions of Discrete Distributions
[abstract]
[pdf]
C. Canonne, I. Diakonikolas, T. Gouleakis, R. Rubinfeld
Proceedings of the 33rd International Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Invited to Special Issue for STACS 2016
We study the question of testing
structured properties (classes) of discrete distributions. Specifically, given
sample access to an arbitrary distribution $D$ over $[n]$ and a property $\mathcal{P}$, the goal is to distinguish
between $D\in \mathcal{P}$ and $D$ is $\epsilon$far in $\ell_1$ distance from $\mathcal{P}$.
We develop a general algorithm for this question,
which applies to a large range of "shapeconstrained" properties,
including monotone, logconcave, $t$modal, piecewisepolynomial, and Poisson Binomial distributions.
Moreover, for all cases considered, our algorithm has nearoptimal sample complexity
with regard to the domain size and is computationally efficient.
For most of these classes, we provide the first nontrivial tester in the literature.
In addition, we also describe a generic method to prove lower bounds for this problem,
and use it to show our upper bounds are nearly tight.
Finally, we extend some of our techniques to tolerant testing,
deriving nearlytight upper and lower bounds for the corresponding questions.

Differentially Private Learning of Structured Discrete Distributions
[abstract]
[pdf]
[code]
I. Diakonikolas, M. Hardt, L. Schmidt
Advances in Neural Information Processing Systems (NIPS 2015)
We investigate the problem of learning an unknown probability distribution
over a discrete population from random samples. Our goal is to design
efficient algorithms that simultaneously achieve low error in total variation
norm while guaranteeing Differential Privacy to the individuals of the
population.
We describe a general approach that yields near sampleoptimal and computationally efficient differentially
private estimators for a wide range of wellstudied and natural distribution families. Our theoretical results
show that for a wide variety of structured distributions there exist private estimation algorithms that are nearly
as efficientboth in terms of sample size and running timeas their nonprivate counterparts. We complement our theoretical
guarantees with an experimental evaluation. Our experiments illustrate the speed and accuracy
of our private estimators on both synthetic mixture models, as well as a large public data set.

Optimal Algorithms and Lower Bounds for Testing Closeness of Structured Distributions
[abstract]
[pdf]
I. Diakonikolas, D. Kane, V. Nikishkin
Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015)
We give a general unified method that can be used for $L_1$ closeness testing
of a wide range of univariate structured distribution families.
More specifically, we design a sample optimal and computationally efficient algorithm for testing
the identity of two unknown (potentially arbitrary) univariate distributions under the $\mathcal{A}_k$distance metric:
Given sample access to distributions with density functions $p, q: I \to \mathbb{R}$, we want to distinguish
between the cases that $p=q$ and $\pq\_{\mathcal{A}_k} \ge \epsilon$ with probability at least $2/3$.
We show that for any $k \ge 2, \epsilon>0$, the optimal sample complexity of the $\mathcal{A}_k$closeness testing
problem is $\Theta(\max\{ k^{4/5}/\epsilon^{6/5}, k^{1/2}/\epsilon^2 \})$.
This is the first $o(k)$ sample algorithm for this problem, and yields
new, simple $L_1$ closeness testers, in most cases with optimal sample complexity,
for broad classes of structured distributions.

On the Complexity of Optimal Lottery Pricing and Randomized Mechanisms
[abstract]
[pdf]
X. Chen, I. Diakonikolas, A. Orfanou, D. Paparas, X. Sun, M. Yannakakis
Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015)
We study the optimal lottery problem and the optimal mechanism design problem
in the setting of a single unitdemand buyer with item values drawn from independent distributions.
Optimal solutions to both problems are characterized by a linear program with exponentially many variables.
For the menu size complexity of the optimal lottery problem, we present an explicit, simple instance
with distributions of support size $2$, and show that exponentially many lotteries
are required to achieve the optimal revenue. We also show that, when distributions have support size $2$
and share the same high value, the simpler scheme of item pricing can achieve the same revenue as the optimal
menu of lotteries. The same holds for the case of two items with support size $2$
(but not necessarily the same high value).
For the computational complexity of the optimal mechanism design problem,
we show that unless the polynomialtime hierarchy collapses
(more precisely, $\mathrm{P}^{\mathrm{NP}}=\mathrm{P}^{\mathrm{\#P}}$), there is
no universal efficient randomized algorithm to implement
an optimal mechanism even when distributions have support size $3$.

Fast and NearOptimal Algorithms for Approximating Distributions by Histograms
[abstract]
[pdf]
J. Acharya, I. Diakonikolas, C. Hegde, J. Li, L. Schmidt
Proceedings of the 34th Annual ACM Symposium on Principles of Database Systems (PODS 2015)
Histograms are among the most popular structures for the succinct summarization of data in a variety of database applications.
In this work, we provide fast and nearoptimal algorithms for approximating arbitrary
one dimensional data distributions by histograms.
A $k$histogram is a piecewise constant function with $k$ pieces.
We consider the following natural problem, previously studied by Indyk, Levi, and Rubinfeld in PODS 2012:
Given samples from a distribution $p$ over $\{1, \ldots, n \}$, compute a $k$histogram that minimizes
the $\ell_2$distance from $p$, up to an additive $\epsilon$.
We design an algorithm for this problem that uses the informationtheoretically minimal sample
size of $m = O(1/\epsilon^2)$, runs in samplelinear time $O(m)$,
and outputs an $O(k)$ histogram whose $\ell_2$distance from $p$ is at most $O(\mathrm{opt}_k) +\epsilon$,
where $\mathrm{opt}_k$ is the minimum $\ell_2$distance between $p$ and any $k$histogram.
Perhaps surprisingly, the sample size and running time of our algorithm are independent of the universe size $n$.
We generalize our approach to obtain fast algorithms for multiscale histogram construction,
as well as approximation by piecewise polynomial distributions.
We experimentally demonstrate one to two orders of magnitude improvement in terms of empirical
running times over previous stateoftheart algorithms.

Testing Identity of Structured Distributions
[abstract]
[pdf]
I. Diakonikolas, D. Kane, V. Nikishkin
Proceedings of the 26th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2015)
We study the question of identity testing for structured distributions.
More precisely, given samples from a {\em structured} distribution $q$ over $[n]$ and an explicit distribution $p$ over $[n]$,
we wish to distinguish whether $q=p$ versus $q$ is at least $\epsilon$far from $p$,
in $L_1$ distance. In this work, we present a unified approach that yields new, simple testers, with sample complexity
that is informationtheoretically optimal, for broad classes of structured distributions, including $t$flat distributions,
$t$modal distributions, logconcave distributions, monotone hazard rate (MHR) distributions, and mixtures thereof.

Learning from Satisfying Assignments
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Proceedings of the 26th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2015)
This paper studies the problem of learning ``lowcomplexity"
probability distributions over the Boolean hypercube $\{1,1\}^n$.
As in the standard PAC learning model, a learning problem in our
framework is defined by a class ${\cal C}$ of Boolean functions
over $\{1,1\}^n$, but
in our model the learning algorithm is given uniform random satisfying
assignments of an unknown $f \in \cal{C}$ and its goal is to output
a highaccuracy approximation of the uniform distribution
over $f^{1}(1).$ This distribution learning problem may be viewed as a
demanding variant of standard Boolean function learning, where the learning
algorithm only receives positive examples and  more importantly 
must output a hypothesis function which has small \emph{multiplicative}
error (i.e., small error relative to the size of $f^{1}(1)$).
As our main results, we show that the two most widely studied
classes of Boolean functions in computational learning theory 
linear threshold functions and DNF formulas  have
efficient distribution learning algorithms in our model.
Our algorithm for linear threshold functions runs in
time poly$(n,1/\epsilon)$ and our algorithm for
polynomialsize DNF runs in time quasipoly$(n,1/\epsilon)$.
We obtain both these results via a general approach that
combines a broad range of technical ingredients, including the complexitytheoretic study
of approximate counting and uniform generation;
the Statistical Query model from learning theory; and hypothesis testing techniques from statistics.
A key conceptual and technical ingredient of
this approach is a new kind of algorithm which we devise called a ``densifier'' and which we
believe may be useful in other contexts.
We also establish limitations on efficient learnability in our model by showing
that the existence of certain types of cryptographic signature schemes
imply that certain learning problems in our framework are computationally
hard. Via this connection we show that
assuming the existence of sufficiently strong unique signature schemes,
there are no subexponential time learning algorithms in our framework for
intersections of two halfspaces, for
degree2 polynomial threshold functions, or for monotone 2CNF formulas.
Thus our positive results for distribution learning come close to the
limits of what can be achieved by efficient algorithms.

NearOptimal Density Estimation in NearLinear Time Using VariableWidth Histograms
[abstract]
[pdf]
S. Chan, I. Diakonikolas, R. Servedio, X. Sun
Advances in Neural Information Processing Systems (NIPS 2014)
Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We consider the problem
of density estimation, in which a learning algorithm is given i.i.d. draws from $p$ and must
(with high probability) output a hypothesis distribution that is close to $p$. The main contribution of this paper is
a highly efficient density estimation algorithm for learning using a variablewidth histogram, i.e., a
hypothesis distribution with a piecewise constant probability density function.
In more detail, for any $k$ and $\epsilon$, we give an algorithm that makes $\tilde{O}(k/\epsilon^2)$ draws from $p$, runs in $\tilde{O}(k/\epsilon^2)$ time, and outputs a
hypothesis distribution $h$ that is piecewise constant with $O(k \log^2(1/\epsilon))$ pieces. With high probability the
hypothesis $h$ satisfies $d_{\mathrm{TV}}(p,h) \leq C \cdot \mathrm{opt}_k(p) + \epsilon$, where $d_{\mathrm{TV}}$ denotes the total variation distance
(statistical distance), $C$ is a universal constant,
and $\mathrm{opt}_k(p)$ is the smallest total variation distance between $p$ and any $k$piecewise constant distribution.
The sample size and running time of our algorithm are optimal up to logarithmic factors.
The ``approximation factor'' $C$ in our result is inherent in the problem, as we prove that
no algorithm with sample size bounded in terms of $k$ and $\epsilon$ can achieve $C<2$ regardless
of what kind of hypothesis distribution it uses.

Efficient Density Estimation via Piecewise Polynomial Approximation
[abstract]
[pdf]
S. Chan, I. Diakonikolas, R. Servedio, X. Sun
Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014)
We give a highly efficient "semiagnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $\tau$close (in total variation distance) to an unknown probability distribution $q$ that is defined by an unknown partition of $I$ into $t$ intervals and $t$ unknown degree$d$ polynomials specifying $q$ over each of the intervals. We give an algorithm that draws $\tilde{O}(t(d+1)/\epsilon^2)$ samples from $p$, runs in time $\mathrm{poly}(t,d,1/\epsilon)$, and with high probability outputs a piecewise polynomial hypothesis distribution $h$ that is $(O(\tau)+\epsilon)$close (in total variation distance) to $p$. This sample complexity is essentially optimal; we show that even for $\tau=0$, any algorithm that learns an unknown $t$piecewise degree$d$ probability distribution over $I$ to accuracy $\epsilon$ must use $\Omega({\frac {t(d+1)} {\mathrm{poly}(1 + \log(d+1))}} \cdot {\frac 1 {\epsilon^2}})$ samples from the distribution, regardless of its running time. Our algorithm combines tools from approximation theory, uniform convergence, linear programming, and dynamic programming.
We apply this general algorithm to obtain a wide range of results for many natural problems in density estimation over both continuous and discrete domains. These include stateoftheart results for learning mixtures of logconcave distributions; mixtures of $t$modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of $k$monotone densities. Our general technique yields computationally efficient algorithms for all these problems, in many cases with provably optimal sample complexities (up to logarithmic factors) in all parameters.

Deterministic Approximate Counting for Juntas of Degree$2$ Polynomial Threshold Functions
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC 2014)
Let $g: \{1,1\}^k \rightarrow \{1,1\}$ be any Boolean function
and $q_1,\dots,q_k$ be any degree$2$ polynomials over
$\{1,1\}^n.$ We give a
deterministic algorithm which,
given as input explicit descriptions of
$g,q_1,\dots,q_k$ and an accuracy parameter $\epsilon>0$,
approximates
\[
\mathbf{Pr}_{x \sim \{1,1\}^n}[g(\mathrm{sign}(q_1(x)),\dots,\mathrm{sign}(q_k(x)))=1]
\]
to within an additive $\pm \epsilon$. For any constant $\epsilon > 0$
and $k \geq 1$ the running time of our algorithm is a fixed
polynomial in $n$ (in fact this is true even for some nottoosmall
$\epsilon = o_n(1)$ and nottoolarge $k = \omega_n(1)$).
This is the first fixed polynomialtime algorithm
that can deterministically approximately count
satisfying assignments of a natural
class of depth$3$ Boolean circuits.
Our algorithm extends a recent result \cite{DDS13:deg2count}
which gave a deterministic
approximate counting algorithm for a single degree$2$ polynomial
threshold function $\mathrm{sign}(q(x)),$ corresponding to the $k=1$ case of our
result. Note that even in the $k=1$ case it is NPhard to determine
whether $\mathbf{Pr}_{x \sim \{1,1\}^n}[\mathrm{sign}(q(x))=1]$ is nonzero,
so any sort of multiplicative approximation is almost certainly
impossible even for efficient randomized algorithms.
Our algorithm and analysis requires several novel technical ingredients
that go significantly beyond the tools required to handle the $k=1$ case
in \cite{DDS13:deg2count}. One of these
is a new multidimensional central limit theorem
for degree$2$ polynomials in Gaussian random variables which builds
on recent Malliavincalculusbased results from probability theory. We
use this CLT as the basis of a new decomposition technique for $k$tuples
of degree$2$ Gaussian polynomials and thus obtain an efficient
deterministic approximate counting
algorithm for the Gaussian distribution, i.e., an algorithm for estimating
\[
\mathbf{Pr}_{x \sim \mathcal{N}(0,1)^n}[g(\mathrm{sign}(q_1(x)),\dots,\mathrm{sign}(q_k(x)))=1].
\]
Finally, a third new ingredient
is a ``regularity lemma'' for $k$tuples of degree$d$
polynomial threshold functions. This generalizes both the regularity lemmas
of \cite{DSTW:10,HKM:09}
(which apply to a single degree$d$ polynomial threshold
function) and the regularity lemma of Gopalan et al \cite{GOWZ10}
(which applies to
a $k$tuples of linear threshold functions, i.e., the case $d=1$).
Our new regularity lemma lets us extend our deterministic approximate
counting results from the Gaussian to the Boolean domain.

The Complexity of Optimal Multidimensional Pricing
[abstract]
[pdf]
X. Chen, I. Diakonikolas, D. Paparas, X. Sun, M. Yannakakis
Games and Economic Behavior, accepted with minor revisions
Proceedings of the 25th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2014)
We resolve the complexity of revenueoptimal deterministic auctions in
the unitdemand singlebuyer Bayesian setting, i.e., the optimal item
pricing problem, when the buyer's values for the items are independent.
We show that the problem of computing a revenueoptimal pricing can be solved
in polynomial time for distributions of support size $2$
and its decision version is NPcomplete for distributions of support size $3$.
We also show that the problem remains NPcomplete for the case of identical distributions.

Optimal Algorithms for Testing Closeness of Discrete Distributions
[abstract]
[pdf]
S. Chan, I. Diakonikolas, G. Valiant, P. Valiant
Proceedings of the 25th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2014)
Blog post about this work: Property Testing Review
We study the question of closeness testing for two discrete distributions.
More precisely, given samples from two distributions $p$ and $q$ over an $n$element set,
we wish to distinguish whether $p=q$ versus $p$ is at least $\epsilon$far from $q$,
in either $\ell_1$ or $\ell_2$ distance. Batu et al~\cite{BFR+:00, Batu13} gave the first sublinear time algorithms for these problems,
which matched the lower bounds of~\cite{PV11sicomp} up to a logarithmic factor in $n$, and a polynomial factor of $\epsilon.$
In this work, we present simple testers for both the $\ell_1$ and $\ell_2$ settings, with sample complexity
that is informationtheoretically optimal, to constant factors, both in the dependence on $n$, and the dependence on $\epsilon$;
for the $\ell_1$ testing problem we establish that the sample complexity is $\Theta(\max\{n^{2/3}/\epsilon^{4/3}, n^{1/2}/\epsilon^2 \}).$

A Polynomialtime Approximation Scheme for Faulttolerant Distributed Storage
[abstract]
[pdf]
C. Daskalakis, A. De, I. Diakonikolas, A. Moitra, R. Servedio
Proceedings of the 25th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2014)
Featured in Abstract Talk. Podcast is here
We consider a problem which has received considerable attention in
systems literature because of its applications to routing in delay tolerant networks and replica placement
in distributed storage systems.
In abstract terms the problem can be stated as follows: Given a random variable $X$
generated by a known product distribution over $\{0,1\}^n$ and a target
value $0 \leq \theta \leq 1$, output a nonnegative vector $w$, with
$\w\_1 \le 1$, which maximizes the probability of the event $w \cdot X
\ge \theta$. This is a challenging nonconvex optimization problem for
which even computing the value $\Pr[w \cdot X \ge \theta]$ of a proposed
solution vector $w$ is #Phard.
We provide an additive EPTAS for this problem
which, for constantbounded product distributions,
runs in $ \mathrm{poly}(n) \cdot 2^{\mathrm{poly}(1/\epsilon)}$ time and outputs an
$\epsilon$approximately optimal solution vector $w$ for this problem. Our
approach is inspired by, and extends,
recent structural results from the complexitytheoretic
study of linear threshold functions. Furthermore, in spite of the objective function being nonsmooth,
we give a unicriterion PTAS while previous work for such objective functions has typically
led to a bicriterion PTAS. We believe our techniques may be applicable to get unicriterion PTAS for other nonsmooth objective functions.

Learning Sums of Independent Integer Random Variables
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. O'Donnell, R. Servedio, LY. Tan
Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013)
Blog post about this work: MIT theory student blog
Let $\mathbf{S} = \mathbf{X}_1 + \cdots + \mathbf{X}_n$ be a sum of $n$ independent
integer random variables $\mathbf{X}_i$, where each $\mathbf{X}_i$ is supported on
$\{0,1,\dots,k1\}$ but otherwise may have an arbitrary distribution
(in particular the $\mathbf{X}_i$'s need not be identically distributed).
How many samples are required to learn the distribution $\mathbf{S}$ to high
accuracy? In this paper we show
that the answer is
completely independent of $n$, and moreover we give a
computationally efficient algorithm which achieves this low sample
complexity. More precisely, our algorithm learns any such $\mathbf{S}$ to $\epsilon$accuracy (with respect
to the total variation distance between distributions)
using $\mathrm{poly}(k,1/\epsilon)$ samples, independent of $n$. Its running time is
$\mathrm{poly}(k,1/\epsilon)$ in the standard word RAM model. Thus we give
a broad generalization of the main result of~\cite{DDS12stoc}
which gave a similar learning result for the special case $k=2$ (when
the distribution~$\mathbf{S}$ is a Poisson Binomial Distribution).
Prior to this work, no nontrivial results were
known for learning these distributions even in the case $k=3$.
A key difficulty is that, in contrast to the case of $k = 2$,
sums of independent $\{0,1,2\}$valued random variables may
behave very differently from
(discretized) normal distributions, and in fact may be
rather complicated  they are not logconcave, they can
be $\Theta(n)$modal, there is no relationship between
Kolmogorov distance and total variation distance for the class, etc.
Nevertheless, the heart of our learning result is a new limit theorem
which characterizes what the sum of an arbitrary number of arbitrary
independent $\{0,1,\dots, k1\}$valued random variables may look like.
Previous limit theorems in this setting made strong assumptions on the
"shift invariance" of the random variables $\mathbf{X}_i$ in order to
force a discretized normal limit. We believe that our new
limit theorem, as the first result for truly arbitrary sums of
independent $\{0,1,\dots,k1\}$valued random variables,
is of independent interest.

Deterministic Approximate Counting for Degree$2$ Polynomial Threshold Functions
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Manuscript, 2013. Available as ECCC technical report [link]
We give a
deterministic algorithm for
approximately computing the fraction of Boolean assignments
that satisfy a degree$2$ polynomial threshold function.
Given a degree$2$ input polynomial $p(x_1,\dots,x_n)$
and a parameter $\epsilon > 0$, the algorithm approximates
$\Pr_{x \sim \{1,1\}^n}[p(x) \geq 0]$
to within an additive $\pm \epsilon$ in time $\mathrm{poly}(n,2^{\mathrm{poly}(1/\epsilon)})$.
Note that it is NPhard to determine whether the above probability
is nonzero, so any sort of multiplicative approximation is almost certainly
impossible even for efficient randomized algorithms.
This is the first deterministic algorithm for this counting problem
in which the running time is polynomial in $n$ for $\epsilon= o(1)$.
For "regular" polynomials $p$ (those in which no individual variable's
influence is large compared to the sum of all $n$
variable influences)
our algorithm runs in $\mathrm{poly}(n,1/\epsilon)$ time.
The algorithm also runs in $\mathrm{poly}(n,1/\epsilon)$ time to approximate
$\Pr_{x \sim \mathcal{N}(0,1)^n}[p(x) \geq 0]$ to within an additive $\pm \epsilon$,
for any degree2 polynomial $p$.
As an application of our counting result, we give a deterministic
multiplicative $(1 \pm \epsilon)$approximation algorithm
to approximate the $k$th absolute moment $\mathbf{E}_{x \sim \{1,1\}^n}[p(x)^k]$
of a degree$2$ polynomial. The algorithm runs in fixed
polynomial time for any constants $k$ and $\epsilon.$

A robust Khintchine Inequality and computing optimal constants in Fourier analysis and highdimensional geometry
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
SIAM Journal on Discrete Mathematics, 302 (2016), pp. 10581094
Proceedings of the 40th Intl. Colloquium on Automata, Languages and Programming (ICALP 2013)
This paper makes two contributions towards determining some wellstudied
optimal constants in Fourier analysis of Boolean functions
and highdimensional geometry.
(1) It has been known since 1994 \cite{GL:94} that every linear threshold function has squared Fourier mass
at least $1/2$ on its degree$0$ and degree$1$ coefficients.
Denote the minimum such Fourier mass by $\mathbf{W}^{\leq 1}[\mathbf{LTF}]$,
where the minimum is taken over all $n$variable linear threshold functions and all $n \ge 0$.
Benjamini, Kalai and Schramm \cite{BKS:99}
have conjectured that the true value of $\mathbf{W}^{\leq 1}[\mathbf{LTF}]$ is $2/\pi$.
We make progress on this conjecture by proving that $\mathbf{W}^{\leq 1}[\mathbf{LTF}]
\geq 1/2 + c$ for some absolute constant $c>0$.
The key ingredient in our proof is a "robust" version of the wellknown
Khintchine inequality in functional analysis, which we
believe may be of independent interest.
(2) We give an algorithm with the following property: given any $\eta > 0$,
the algorithm runs in time $2^{\mathrm{poly}(1/\eta)}$ and determines the value of
$\mathbf{W}^{\leq 1}[\mathbf{LTF}]$ up to an additive error of $\pm\eta$. We give a similar
$2^{{\mathrm{poly}(1/\eta)}}$time algorithm to determine Tomaszewski's constant
to within an additive error of $\pm \eta$; this is the
minimum (over all origincentered hyperplanes $H$) fraction of points
in $\{1,1\}^n$ that lie within Euclidean distance $1$ of $H$.
Tomaszewski's constant is conjectured to be $1/2$; lower bounds on it
have been given by Holzman and Kleitman \cite{HK92} and
independently by BenTal, Nemirovski and Roos
\cite{BNR02}.
Our algorithms combine tools from anticoncentration
of sums of independent random variables, Fourier analysis, and Hermite
analysis of linear threshold functions.

Learning Mixtures of Structured Distributions over Discrete Domains
[abstract]
[pdf]
S. Chan, I. Diakonikolas, R. Servedio, X. Sun
Proceedings of the 24th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2013)
Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n] = \{1,\dots,n\}.$
We show that if $\mathfrak{C}$ satisfies a rather general condition  essentially, that each distribution in
$\mathfrak{C}$ can be wellapproximated by a variablewidth
histogram with few bins  then there is a highly efficient (both in terms of running time and sample complexity)
algorithm that can learn any mixture of $k$ unknown distributions from
$\mathfrak{C}.$
We analyze several natural types of distributions over $[n]$,
including logconcave, monotone hazard rate and unimodal distributions,
and show that they have the required structural property of being
wellapproximated by a histogram with few bins.
Applying our general algorithm, we
obtain nearoptimally efficient algorithms for all these mixture
learning problems as described below. More precisely,
Logconcave distributions: We learn any mixture of $k$
logconcave distributions over $[n]$ using $k \cdot
\tilde{O}(1/\epsilon^4)$ samples (independent of $n$) and running in time
$\tilde{O}(k \log(n) / \epsilon^4)$ bitoperations (note that reading a single
sample from $[n]$ takes $\Theta(\log n)$ bit operations).
For the special case $k=1$ we give an efficient
algorithm using $\tilde{O}(1/\epsilon^3)$
samples; this generalizes the main result of \cite{DDS12stoc} from the
class of Poisson Binomial distributions to the much broader class of all
logconcave distributions. Our upper bounds are not far from
optimal since any algorithm for this learning problem requires
$\Omega(k/\epsilon^{5/2})$ samples.
Monotone hazard rate (MHR) distributions:
We learn any mixture of $k$ MHR distributions over $[n]$ using
$O(k \log (n/\epsilon)/\epsilon^4)$ samples and running in time $\tilde{O}(k
\log^2(n) / \epsilon^4)$ bitoperations. Any algorithm for this learning problem must use $\Omega(k \log(n)/\epsilon^3)$ samples.
Unimodal distributions:
We give an algorithm that learns any mixture of $k$ unimodal distributions
over $[n]$ using $O(k \log (n)/\epsilon^{4})$ samples and running in time
$\tilde{O}(k \log^2(n) / \epsilon^{4})$ bitoperations.
Any algorithm for this problem must use $\Omega(k \log(n)/\epsilon^3)$ samples.

Testing $k$modal Distributions: Optimal Algorithms via Reductions
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. Servedio, G. Valiant, P. Valiant
Proceedings of the 24th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2013)
We give highly efficient algorithms, and almost matching lower bounds, for a range of basic statistical problems
that involve testing and estimating the $L_1$ (total variation) distance between two $k$modal distributions $p$ and $q$ over the discrete domain $\{1,\dots,n\}$.
More precisely, we consider the following four problems: given sample access to an unknown $k$modal
distribution $p$,
Testing identity to a known or unknown distribution:
(1) Determine whether $p = q$ (for an explicitly given $k$modal distribution $q$) versus
$p$ is $\epsilon$far from $q$;
(2) Determine whether $p=q$ (where $q$ is available via sample access) versus
$p$ is $\epsilon$far from $q$;
Estimating $L_1$ distance (``tolerant testing'') against a known or unknown distribution:
(3) Approximate $d_{TV}(p,q)$ to within additive $\epsilon$ where $q$ is an explicitly
given $k$modal distribution $q$;
(4)Approximate $d_{TV}(p,q)$ to within additive $\epsilon$ where $q$ is available via sample access.
For each of these four problems we give sublogarithmic sample algorithms, that we show are tight up to additive $\mathrm{poly}(k)$
and multiplicative $\mathrm{polylog}\log n+\mathrm{polylog} k$ factors.
Thus our bounds significantly improve the previous results of \cite{BKR:04}, which were for testing identity of distributions (items (1) and (2) above) in the special cases
$k=0$ (monotone distributions) and $k=1$ (unimodal distributions) and required $O((\log n)^3)$ samples.

An Optimal Algorithm for the Efficient Approximation of Convex Pareto Curves
I. Diakonikolas, M. Yannakakis
Manuscript, 2012

EfficiencyRevenue Tradeoffs in Auctions
[abstract]
[pdf]
I. Diakonikolas, C.H. Papadimitriou, G. Pierrakos, Y. Singer
Proceedings of the 39th Intl. Colloquium on Automata, Languages and Programming (ICALP 2012)
When agents with independent priors bid for a single item, Myerson's optimal auction maximizes expected revenue, whereas Vickrey's secondprice auction optimizes social welfare. We address the natural question of tradeoffs between the two criteria, that is, auctions that optimize, say, revenue under the constraint that the welfare is above a given level. If one allows for randomized mechanisms, it is easy to see that there are polynomialtime mechanisms that achieve any point in the tradeoff (the Pareto curve) between revenue and welfare. We investigate whether one can achieve the same guarantees using deterministic mechanisms. We provide a negative answer to this question by showing that this is a (weakly) NPhard problem. On the positive side, we provide polynomialtime deterministic mechanisms that approximate with arbitrary precision any point of the tradeoff between these two fundamental objectives for the case of two bidders, even when the valuations are correlated arbitrarily. The major problem left open by our work is whether there is such an algorithm for three or more bidders with independent valuation distributions.

The Inverse Shapley Value Problem
[abstract]
[pdf]
A. De, I. Diakonikolas, R. Servedio
Games and Economic Behavior, accepted with minor revisions
Proceedings of the 39th Intl. Colloquium on Automata, Languages and Programming (ICALP 2012)
For $f$ a weighted voting scheme used by $n$ voters to choose between two
candidates, the $n$
ShapleyShubik Indices (or
Shapley values)
of $f$ provide a measure of how
much control each voter can exert over the overall outcome of the vote.
ShapleyShubik indices were introduced by Lloyd Shapley
and Martin Shubik in 1954 \cite{SS54} and are widely studied in
social choice theory as a measure of the "influence" of voters.
The
Inverse Shapley Value Problem is the problem of designing a weighted
voting scheme which (approximately) achieves a desired input vector of
values for the ShapleyShubik indices. Despite much interest in this problem
no provably correct and efficient algorithm was known prior to our work.
We give the first efficient algorithm with provable performance guarantees for
the Inverse Shapley Value Problem. For any constant $\epsilon > 0$
our algorithm runs in fixed poly$(n)$ time (the degree of the
polynomial is independent of $\epsilon$) and has the following
performance guarantee: given as input a vector of desired Shapley values,
if any "reasonable" weighted voting scheme
(roughly, one in which the threshold is not too skewed)
approximately matches the desired vector of values to within
some small error,
then our algorithm explicitly outputs a weighted voting scheme that
achieves this vector of Shapley values to within error $\epsilon.$
If there is a "reasonable" voting scheme in which all
voting weights are integers at most $\mathrm{poly}(n)$ that approximately achieves
the desired Shapley values, then our algorithm runs in time
$\mathrm{poly}(n)$ and outputs a weighted voting scheme that achieves
the target vector of Shapley values to within
error $\epsilon=n^{1/8}.$

Nearly optimal solutions for the Chow Parameters Problem and lowweight approximation of halfspaces
[abstract]
[pdf]
A. De, I. Diakonikolas, V. Feldman, R. Servedio
Journal of the ACM, 61(2), 2014. Invited to Theory of Computing special issue on Analysis of Boolean functions (declined)
Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012)
IBM Research 2014 Pat Goldberg Math/CS/EE Best Paper Award
The
Chow parameters of a Boolean function $f: \{1,1\}^n \to \{1,1\}$ are its $n+1$ degree$0$ and
degree$1$ Fourier coefficients. It has been known since 1961 \cite{Chow:61, Tannenbaum:61} that the (exact values of the) Chow parameters of
any linear threshold function $f$ uniquely specify $f$ within the space of all Boolean functions, but until
recently \cite{OS11:chow} nothing was known about efficient algorithms for
reconstructing $f$
(exactly or approximately) from exact or approximate values of its Chow parameters. We refer to this reconstruction problem as the
Chow Parameters Problem.
Our main result is a new algorithm for the Chow Parameters Problem which,
given (sufficiently accurate approximations to) the Chow parameters of any linear threshold function $f$, runs in time
$\tilde{O}(n^2)\cdot (1/\epsilon)^{O(\log^2(1/\epsilon))}$ and
with high probability outputs a representation of an LTF $f'$ that is $\epsilon$close to $f$ in Hamming distance.
The only previous algorithm \cite{OS11:chow} had running time $\mathrm{poly}(n) \cdot 2^{2^{\tilde{O}(1/\epsilon^2)}}.$
As a byproduct of our approach, we show that for any linear threshold function $f$ over $\{1,1\}^n$,
there is a linear threshold function $f'$ which is $\epsilon$close to $f$ and has all weights that are integers of magnitude at most $\sqrt{n} \cdot (1/\epsilon)^{O(\log^2(1/\epsilon))}$.
This significantly improves the previous best result of~\cite{DiakonikolasServedio:09} which gave
a $\mathrm{poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}$ weight bound, and is close to the
known lower bound of
$\max\{\sqrt{n},$ $(1/\epsilon)^{\Omega(\log \log (1/\epsilon))}\}$ \cite{Goldberg:06b,Servedio:07cc}.
Our techniques also yield improved algorithms for related problems in learning theory.
In addition to being significantly stronger than previous work, our results
are obtained using conceptually simpler proofs.
The two main ingredients underlying our results are (1) a new structural
result showing that for $f$ any linear threshold function and $g$ any bounded
function, if the Chow parameters of $f$ are close to the Chow
parameters of $g$ then $f$ is close to $g$; (2) a new boostinglike algorithm
that given approximations to the Chow parameters of a linear threshold function outputs a bounded
function whose Chow parameters are close to those of $f$.

Learning Poisson Binomial distributions
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. Servedio
Invited to Algorithmica special issue on New Theoretical Challenges in Machine Learning
Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012)
We consider a basic problem in unsupervised learning:
learning an unknown
Poisson Binomial Distribution.
A Poisson Binomial Distribution (PBD) over $\{0,1,\dots,n\}$
is the distribution of a sum of $n$ independent Bernoulli
random variables which may have arbitrary, potentially nonequal,
expectations. These distributions were first studied by S. Poisson in 1837 \cite{Poisson:37} and are a natural
$n$parameter generalization of the familiar Binomial Distribution.
Surprisingly, prior to our work this basic learning problem
was poorly understood, and known results for it were far from
optimal.
We essentially settle the complexity of the learning problem for this
basic class of distributions.
As our main result we give a highly efficient algorithm which learns to
$\epsilon$accuracy using $\tilde{O}(1/\epsilon^3)$ samples independent of $n$.
The running time of the algorithm is quasilinear in the
size of its input data. This is nearly optimal
since any algorithm must use $\Omega(1/\epsilon^2)$ samples.
We also give positive and negative results for some extensions of this learning problem.

Learning $k$modal distributions via testing
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, R. Servedio
Theory of Computing, 10 (20), 535570 (2014)
Proceedings of the 23rd Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2012)
Oded Goldreich's Choices #72
A $k$modal probability distribution over the domain $\{1,...,n\}$ is one whose
histogram has at most $k$ "peaks" and "valleys." Such distributions are
natural generalizations of monotone ($k=0$) and unimodal ($k=1$)
probability distributions, which have been intensively studied in probability theory and statistics.
In this paper we consider the problem of learning an unknown $k$modal distribution.
The learning algorithm is given access to independent samples drawn from the $k$modal
distribution $p$, and must output a hypothesis distribution $\hat{p}$ such that with high
probability the total variation distance between $p$ and $\hat{p}$ is at most $\epsilon.$
We give an efficient algorithm for this problem that runs in time $\mathrm{poly}(k,\log(n),1/\epsilon)$.
For $k \leq \tilde{O}(\sqrt{\log n})$, the number of samples used by our algorithm is very close (within an
$\tilde{O}(\log(1/\epsilon))$ factor) to being informationtheoretically optimal. Prior to this
work computationally efficient algorithms were known only for the cases $k=0,1$
\cite{Birge:87b,Birge:97}.
A novel feature of our approach is that our learning algorithm crucially uses a new property testing algorithm as a key subroutine.
The learning algorithm uses the property tester to efficiently
decompose the $k$modal distribution into $k$ (near)monotone distributions, which are easier to
learn.

Noise Stable Halfspaces are Close to Very Small Juntas
[abstract]
[pdf]
I. Diakonikolas, R. Jaiswal, R. Servedio, L.Y.Tan, A. Wan
Chicago Journal of Theoretical Computer Science, 2015
Bourgain~\cite{Bourgain:02} showed that any noise stable Boolean function $f$
can be wellapproximated by a junta.
In this note we give an exponential sharpening of the parameters of
Bourgain's result under the additional assumption that $f$ is a halfspace.

Supervised Design Space Exploration by Compositional Approximation of Pareto sets
[abstract]
[pdf]
H.Y. Liu, I. Diakonikolas, M. Petracca, L.P. Carloni
Proceedings of the 48th Design Automation Conference (DAC 2011)
Technology scaling allows the integration of billions of transistors on the same
die but CAD tools struggle in keeping up with the increasing design complexity.
Design productivity for multicore SoCs increasingly depends on
creating and maintaining reusable components and hierarchically
combining them to form larger composite cores.
Characterizing such composite cores with respect to their power/performance
tradeoffs is critical for design reuse across various products and relies
heavily on synthesis tools.
We present $\mathrm{CAPS}$, an online adaptive algorithm that efficiently
explores the design space of any given core and returns an accurate
characterization of its implementation tradeoffs in terms of an approximate
Pareto set.
It does so by supervising the order of the timeconsuming logicsynthesis runs
on the core's components.
Our algorithm can provably achieve the desired precision on the approximation in
the shortest possible time, without having any apriori information on any
component.
We also show that, in practice, $\mathrm{CAPS}$ works even better than what is guaranteed
by the theory.

DisjointPath Facility Location: Theory and Practice
[abstract]
[pdf]
L. Breslau, I. Diakonikolas, N. Duffield, Y.Gu, M.T. Hajiaghayi, D.S. Johnson, H. Karloff, M. Resende, S.Sen
Proceedings of the 13th Workshop on Algorithm Engineering and Experiments (ALENEX 2011)
Journal version in Operations Research, 2019 [arxiv]
This paper is a theoretical and experimental study of two related facility
location problems that emanated from networking. Suppose we are given a
network modeled as a directed graph $G = (V,A)$, together with
(notnecessarilydisjoint) subsets $C$ and $F$ of $V$ , where $C$ is a set of
customer locations and $F$ is a set of potential facility locations (and
typically $C\subseteq F$). Our goal is to find a minimum sized subset $F' \subseteq F$
such that for every customer $c \in C$ there are two locations $f_1, f_2 \in F'$
such that traffic from $c$ to $f_1$ and to $f_2$ is routed on disjoint paths
(usually shortest paths) under the network's routing protocols.
Although we prove that this problem is impossible to approximate in the
worst case even to within a factor of $2^{\log^{1\epsilon} n}$ for any $\epsilon>0$
(assuming no NPcomplete language can be solved in quasipolynomial
time), we show that the situation is much better in practice. We
propose three algorithms that build solutions and determine lower
bounds on the optimum solution, and evaluate them on several large real
ISP topologies and on synthetic networks designed to reflect realworld
LAN/WAN network structure. Our main algorithms are (1) an algorithm
that performs multiple runs of a straightforward randomized greedy
heuristic and returns the best result found, (2) a genetic
algorithm that uses the greedy algorithm as a subroutine, and (3) a new
"Double Hitting Set" algorithm. All three approaches perform surprising
well, although, in practice, the most costeffective approach is the
multirun greedy algorithm. This yields results that average within 0.7%
of optimal for our synthetic instances and within 2.9% for our
realworld instances, excluding the largest (and most realistic) one.
For the latter instance, the other two algorithms come into their own,
finding solutions that are more than three times better than those of
the multistart greedy approach. In terms of our motivating monitoring
application, where every customer location can be a facility location,
the results are even better. Here the above Double Hitting Set solution
is 90% better than the default solution which places a monitor at each
customer location. Our results also show that, on
average for our realworld instances, we could save an additional 18%
by choosing the (shortest path) routes ourselves, rather than taking
the simpler approach of relying on the network to choose them for us.

Hardness Results for Agnostically Learning LowDegree Polynomial Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, R. O'Donnell, R. Servedio, Y.Wu
Proceedings of the 22nd Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2011)
Hardness results for maximum agreement problems have close connections to hardness results for
proper learning in computational learning theory.
In this paper we prove two hardness results for the problem of finding a low degree polynomial threshold function (PTF)
which has the maximum possible agreement with a given set of labeled examples in $\mathbf{R}^n \times \{1,1\}.$
We prove that for any constants $d\geq 1, \epsilon > 0$,
(1) Assuming the Unique Games Conjecture, no polynomialtime algorithm can find a degree$d$ PTF that
is consistent with a $(1/2 + \epsilon)$ fraction of a given set of labeled examples in $\mathbf{R}^n \times \{1,1\}$,
even if there exists a degree$d$ PTF that is consistent with a $1\epsilon$ fraction of the examples.
(2) It is NPhard to find a degree$2$ PTF that is consistent with
a $(1/2 + \epsilon)$ fraction of a given set of labeled examples in $\mathbf{R}^n \times \{1,1\}$, even if
there exists a halfspace (degree$1$ PTF) that is consistent with a $1  \epsilon$ fraction of the
examples.
These results immediately imply the following hardness of learning results: (i) Assuming the
Unique Games Conjecture, there is no betterthantrivial proper learning algorithm that agnostically learns degree$d$ PTFs under arbitrary distributions;
(ii) There is no betterthantrivial learning algorithm that outputs degree$2$ PTFs and agnostically learns halfspaces (i.e., degree$1$ PTFs) under arbitrary distributions.

Bounded Independence Fools Degree$2$ Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, D. Kane, J. Nelson
Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010)
Let $x$ be a random vector coming from any $k$wise independent distribution over $\{1,1\}^n$.
For an $n$variate degree$2$ polynomial $p$, we prove that $\mathbf{E}[\mathrm{sgn}(p(x))]$ is determined up to an additive $\epsilon$ for $k =
\mathrm{poly}(1/\epsilon)$. This gives a large class of explicit pseudorandom generators against such functions and answers an open
question of Diakonikolas et al. (FOCS 2009).
In the process, we develop a novel analytic technique we dub multivariate FTmollification. This
provides a generic tool to approximate bounded (multivariate) functions by lowdegree polynomials (with respect to
several different notions of approximation). A univariate version of the method was introduced by Kane et al. (SODA 2010) in
the context of streaming algorithms. In this work, we refine it and generalize it to the multivariate setting. We believe that
our technique is of independent mathematical interest. To illustrate its generality, we note that it implies a multidimensional
generalization of Jackson's classical result in approximation theory due to (Newman and Shapiro, 1963).
To obtain our main result, we combine the FTmollification technique with several linear algebraic and probabilistic
tools. These include the invariance principle of of Mossell, O'Donnell and Oleszkiewicz, anticoncentration bounds for
lowdegree polynomials, an appropriate decomposition of degree$2$ polynomials, and a generalized hypercontractive inequality
for quadratic forms which takes the operator norm of the associated matrix into account. Our analysis is quite modular; it
readily adapts to show that intersections of halfspaces and degree$2$ threshold functions are fooled by bounded independence.
From this it follows that $\Omega(1/\epsilon^2)$wise independence derandomizes the GoemansWilliamson hyperplane rounding scheme.
Our techniques unify, simplify, and in some cases improve several recent results in the literature concerning threshold
functions. For the case of "regular" halfspaces we give a simple proof of an optimal independence bound of
$\Theta(1/\epsilon^2)$, improving upon Diakonikolas et al. (FOCS 2009) by polylogarithmic factors. This yields the first optimal
derandomization of the BerryEsseen theorem and  combined with the results of Kalai et al. (FOCS 2005)  implies a
faster algorithm for the problem of agnostically learning halfspaces.

Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, P. Raghavendra, R. Servedio, L.Y. Tan
SIAM Journal on Computing, 43(1), 231253 (2014)
Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC 2010)
(Conference version merged with this paper by Harsha, Klivans and Meka)
We give the first nontrivial upper bounds on the Boolean average
sensitivity and noise sensitivity of degree$d$ polynomial threshold
functions (PTFs). Our bound on the Boolean average sensitivity of PTFs represents the first progress
towards the resolution of a conjecture of Gotsman and Linial \cite{GL:94}, which states that the symmetric function slicing the
middle $d$ layers of the Boolean hypercube has the highest average
sensitivity of all degree$d$ PTFs. Via the $L_1$ polynomial
regression algorithm of Kalai et al. \cite{KKMS:08}, our bound on
Boolean noise sensitivity yields the first polynomialtime
agnostic learning algorithm for the broad class of constantdegree
PTFs under the uniform distribution.
To obtain our bound on the Boolean average sensitivity of PTFs,
we generalize the "criticalindex" machinery of \cite{Servedio:07cc}
(which in that work applies to halfspaces, i.e., degree$1$ PTFs) to general PTFs.
Together with the "invariance principle" of \cite{MOO10},
this allows us to essentially reduce the Boolean setting
to the Gaussian setting. The main ingredients used to obtain our bound
in the Gaussian setting are tail bounds and anticoncentration bounds on
lowdegree polynomials in Gaussian random variables
\cite{Janson:97,CW:01}. Our bound on Boolean noise sensitivity is achieved
via a simple reduction from upper bounds on average sensitivity of Boolean
PTFs to corresponding bounds on noise sensitivity.

A Regularity Lemma, and Lowweight Approximators, for lowdegree Polynomial Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, R. Servedio, L.Y. Tan, A. Wan
Theory of Computing, 10(2), 2753 (2014)
Proceedings of the 25th Annual IEEE Conference on Computational Complexity (CCC 2010)
We give a "regularity lemma" for degree$d$ polynomial threshold
functions (PTFs) over the Boolean cube $\{1,1\}^n$. Roughly
speaking, this result shows that every degree$d$ PTF can be
decomposed into a constant number of subfunctions such that almost
all of the subfunctions are close to being regular PTFs. Here a "regular" PTF
is a PTF $\mathrm{sign}(p(x))$ where the influence of each variable on the
polynomial $p(x)$ is a small fraction of the total influence of $p.$
As an application of this regularity lemma, we prove that for any constants $d \geq 1, \epsilon > 0$, every degree$d$ PTF over $n$
variables can be approximated to accuracy $\epsilon$ by a constantdegree PTF that has integer weights of total magnitude $O_{\epsilon,d}(n^d).$
This weight bound is shown to be optimal up to logarithmic factors.

How Good is the Chord Algorithm?
[abstract]
[pdf]
C. Daskalakis, I. Diakonikolas, M. Yannakakis
SIAM Journal on Computing, 45(3), pp. 811858, 2016
Proceedings of the 21st Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2010)
The Chord algorithm is a popular, simple method for the succinct approximation of curves,
which is widely used, under different names, in a variety of areas, such as,
multiobjective and parametric optimization, computational geometry, and graphics.
We analyze the performance of the Chord algorithm, as compared to the
optimal approximation that achieves a desired accuracy with the minimum number of points.
We prove sharp upper and lower bounds, both in the worst case and average case setting.

Bounded Independence Fools Halfspaces
[abstract]
[pdf]
I. Diakonikolas, P. Gopalan, R. Jaiswal, R. Servedio, E. Viola
SIAM Journal on Computing, 39(8), 34413462 (2010)
Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009)
We show that any distribution on $\{1, 1\}^n$ that is $k$wise
independent fools any halfspace (a.k.a. linear threshold function) $h : \{1, 1\}^n \to \{1, 1\}$, i.e.,
any function of the form $h(x) = \mathrm{sign}(\sum_{i = 1}^n w_i x_i  \theta)$ where the $w_1,\ldots,w_n,\theta$ are arbitrary real
numbers, with error $\epsilon$ for $k = O(\epsilon^{2}\log^2(1/\epsilon))$. Our result is tight up
to $\log(1/\epsilon)$ factors. Using standard constructions of $k$wise independent distributions, we obtain the first
explicit pseudorandom generators $G : \{1, 1\}^s \to \{1, 1\}^n$ that fool halfspaces.
Specifically, we fool halfspaces with error $\epsilon$ and
seed length $s = k \cdot \log n = O(\log n \cdot \epsilon^{2} \log^2(1/\epsilon))$.
Our approach combines classical tools from real approximation theory with
structural results on halfspaces by Servedio (Comput. Complexity 2007).

Improved Approximation of Linear Threshold Functions
[abstract]
[pdf]
I. Diakonikolas, R. Servedio
Computational Complexity, 22(3), 623677 (2013)
Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC 2009)
We prove two main results on how arbitrary linear threshold
functions $f(x) = \mathrm{sign}(w\cdot x  \theta)$ over the $n$dimensional
Boolean hypercube can be approximated by simple threshold functions.
Our first result shows that every $n$variable threshold function
$f$ is $\epsilon$close to a threshold function depending only on
$\mathrm{Inf}(f)^2 \cdot \mathrm{poly}(1/\epsilon)$ many variables, where $\mathrm{Inf}(f)$
denotes the total influence or average sensitivity of $f.$ This is
an exponential sharpening of Friedgut's wellknown theorem
\cite{Friedgut:98}, which states that every Boolean function $f$ is
$\epsilon$close to a function depending only on $2^{O(\mathrm{Inf}(f)/\epsilon)}$
many variables, for the case of threshold functions. We complement
this upper bound by showing that $\Omega(\mathrm{Inf}(f)^2 + 1/\epsilon^2)$
many variables are required for $\epsilon$approximating threshold
functions.
Our second result is a proof that every $n$variable threshold
function is $\epsilon$close to a threshold function with integer
weights at most $\mathrm{poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.$ This
is an improvement, in the dependence on the error
parameter $\epsilon$, on an earlier result of \cite{Servedio:07cc} which
gave a $\mathrm{poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}$ bound. Our
improvement is obtained via a new proof technique that uses strong
anticoncentration bounds from probability theory. The new technique
also gives a simple and modular proof of the original
\cite{Servedio:07cc} result, and extends to give lowweight
approximators for threshold functions under a range of probability
distributions other than the uniform distribution.

Efficiently Testing Sparse $GF(2)$ Polynomials
[abstract]
[pdf]
I. Diakonikolas, H. Lee, K. Matulef, R. Servedio, A. Wan
Algorithmica, 61(3), 580605 (2011)
Proceedings of the 35th Intl. Colloquium on Automata, Languages and Programming (ICALP 2008)
We give the first algorithm that is both queryefficient and
timeefficient for testing whether an unknown function $f: \{0,1\}^n \to
\{0,1\}$ is an $s$sparse $GF(2)$ polynomial versus $\epsilon$far from
every such polynomial. Our algorithm makes $\mathrm{poly}(s,1/\epsilon)$
blackbox queries to $f$ and runs in time $n \cdot \mathrm{poly}(s,1/\epsilon)$.
The only previous algorithm for this testing problem \cite{DLM+:07}
used $\mathrm{poly}(s,1/\epsilon)$ queries, but had running time exponential in
$s$ and superpolynomial in $1/\epsilon$.
Our approach significantly extends the "testing by implicit
learning" methodology of \cite{DLM+:07}. The learning component of
that earlier work was a bruteforce exhaustive search over a concept
class to find a hypothesis consistent with a sample of random
examples. In this work, the learning component is a
sophisticated exact learning algorithm for sparse $GF(2)$
polynomials due to Schapire and Sellie \cite{SchapireSellie:96}. A
crucial element of this work, which enables us to simulate the
membership queries required by \cite{SchapireSellie:96}, is an
analysis establishing new properties of how sparse $GF(2)$
polynomials simplify under certain restrictions of "lowinfluence"
sets of variables.

Succinct Approximate Convex Pareto Curves
[abstract]
[pdf]
I. Diakonikolas, M. Yannakakis
Proceedings of the 19th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2008)
We study the succinct approximation of convex Pareto curves of multiobjective optimization problems.
We propose the concept of $\epsilon$convex Pareto ($\epsilon$CP) set as the appropriate one for the convex setting, and
observe that it can offer arbitrarily more compact representations than $\epsilon$Pareto sets in this context. We characterize
when an $\epsilon$CP can be constructed in polynomial time in terms of an efficient routine $\textrm{Comb}$ for optimizing
(exactly or approximately) monotone linear combinations of the objectives. We investigate the problem of computing minimum size
$\epsilon$convex Pareto sets, both for discrete (combinatorial) and continuous (convex) problems, and present general
algorithms using a $\textrm{Comb}$ routine. For biobjective problems, we show that if we have an exact $\textrm{Comb}$
optimization routine, then we can compute the minimum $\epsilon$CP for continuous problems (this applies for example to
biobjective Linear Programming and Markov Decision Processes), and factor 2 approximation to the minimum $\epsilon$CP for
discrete problems (this applies for example to biobjective versions of polynomialtime solvable combinatorial problems such as
Shortest Paths, Spanning Tree, etc.). If we have an approximate $\textrm{Comb}$ routine, then we can compute factor 3 and 6
approximations respectively to the minimum $\epsilon$CP for continuous and discrete biobjective problems.
We consider also the case of three and more objectives and present some upper and lower bounds.

Testing for Concise Representations
[abstract]
[pdf]
I. Diakonikolas, H. Lee, K. Matulef, K. Onak, R. Rubinfeld, R. Servedio, A. Wan
Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007)
Oded Goldreich's Choices #39
We describe a general method for testing whether a function on $n$
input variables has a concise representation. The approach combines
ideas from the junta test of Fischer
et al. \cite{FKR+:04} with ideas
from learning theory, and yields property testers that make
poly$(s/\epsilon)$ queries (independent of $n$) for Boolean function
classes such as $s$term DNF formulas
(answering a question posed by Parnas
et al. \cite{PRS02}),
size$s$ decision trees, size$s$ Boolean formulas,
and size$s$ Boolean circuits.
The method can be applied to nonBoolean valued function classes as
well. This is achieved via a generalization of the notion of
variation from Fischer et al. to nonBoolean functions. Using
this generalization we extend the original junta test of Fischer
et al. to work for nonBoolean functions, and give
poly$(s/\epsilon)$query testing algorithms for nonBoolean valued
function classes such as size$s$ algebraic circuits and $s$sparse
polynomials over finite fields.
We also prove an $\tilde\Omega(\sqrt{s})$ query lower bound for
nonadaptively testing $s$sparse polynomials over finite fields of
constant size. This shows that in some instances, our general method
yields a property tester with query complexity that is optimal (for nonadaptive
algorithms) up to a polynomial factor.

Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems
[abstract]
[pdf]
I. Diakonikolas, M. Yannakakis
SIAM Journal on Computing, 39(4), 13401371 (2009)
Proceedings of the 10th Intl. Workshop on Approximation, Randomization, and Combinatorial Optimization (APPROX 2007)
Honorable Mention, George Nicholson student paper competition
INFORMS society, 2009
We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy $\epsilon$ the
Pareto curve of a multiobjective optimization problem. We show that for a broad class of biobjective problems (containing
many important widely studied problems such as shortest paths, spanning tree, matching and many others), we can compute in
polynomial time an $\epsilon$Pareto set that contains at most twice as many solutions as the minimum such set. Furthermore we
show that the factor of $2$ is tight for these problems, i.e., it is NPhard to do better. We present upper and lower bounds
for three or more objectives, as well as for the dual problem of computing a specified number $k$ of solutions which provide a
good approximation to the Pareto curve.