
Robustly Learning any Clusterable Mixture of Gaussians
[abstract]
[arxiv]
I. Diakonikolas, S. Hopkins, D. Kane, S. Karmalkar
Manuscript, 2020
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
Manuscript, 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.

Learning Halfspaces with Massart Noise Under Structured Distributions
[abstract]
[arxiv]
I. Diakonikolas, V. Kontonis, C. Tzamos, N. Zarifis
Manuscript, 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.

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.

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.

Learning Wasserstein distance in spaces of low intrinsic dimension
T. Carpenter, I. Diakonikolas, A. Sidiropoulos
Manuscript

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.