arXiv subjects

Ilias Diakonikolas

Publications and source records attributed to Ilias Diakonikolas.

At least 145 records · Page 8Linked to original sources

arXiv2018

Degree-$d$ Chow Parameters Robustly Determine Degree-$d$ PTFs (and Algorithmic Applications)

The degree-$d$ Chow parameters of a Boolean function $f: \{-1,1\}^n \to \mathbb{R}$ are its degree at most $d$ Fourier coefficients. It is well-known 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 completely independent of the dimension $n$. That is, we show that Boolean degree-$d$ PTFs are {\em robustly identifiable} from their degree-$d$ Chow parameters. Results of this form had been shown for the $d=1$ case~\cite{OS11:chow, DeDFS14}, but no non-trivial 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~\cite{BenDavidDichterman:98}. As a byproduct of our approach, we also obtain the first low integer-weight approximations of degree-$d$ PTFs, for $d>1$. As our second application, our robust identifiability result gives the first efficient algorithm, with dimension-independent error guarantees, for malicious learning of Boolean degree-$d$ PTFs under the uniform distribution.

cs.LG↗

arXiv2018

Robust Learning of Fixed-Structure Bayesian Networks

We investigate the problem of learning Bayesian networks in a robust model where an $ε$-fraction of the samples are adversarially corrupted. In this work, we study the fully observable discrete case where the structure of the network is given. Even in this basic setting, previous learning algorithms either run in exponential time or lose dimension-dependent factors in their error guarantees. We provide the first computationally efficient robust learning algorithm for this problem with dimension-independent error guarantees. Our algorithm has near-optimal sample complexity, runs in polynomial time, and achieves error that scales nearly-linearly with the fraction of adversarially corrupted samples. Finally, we show on both synthetic and semi-synthetic data that our algorithm performs well in practice.

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arXiv2018

Testing Conditional Independence of Discrete Distributions

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 $ε$-far, in $\ell_1$-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 information-theoretic 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 \[ Θ\left({\max\left(n^{1/2}/ε^2,\min\left(n^{7/8}/ε,n^{6/7}/ε^{8/7}\right)\right)}\right)\,. \]

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arXiv2018

Efficient Algorithms and Lower Bounds for Robust Linear Regression

We study the problem of high-dimensional linear regression in a robust model where an $ε$-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 $\mathcal{N}(0, Σ)$ on $\mathbb{R}^d$. 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 near-optimal and computationally efficient algorithm that outputs a candidate hypothesis vector $\widehatβ$ which approximates the unknown regression vector $β$ within $\ell_2$-norm $O(ε\log(1/ε) σ)$, where $σ$ is the standard deviation of the random observation noise. An error of $Ω(εσ)$ is information-theoretically necessary, even with infinite sample size. Prior work gave an algorithm for this problem with sample complexity $\tildeΩ(d^2/ε^2)$ whose error guarantee scales with the $\ell_2$-norm of $β$. For the case of unknown covariance, we show that we can efficiently achieve the same error guarantee as in the known covariance case using an additional $\tilde{O}(d^2/ε^2)$ unlabeled examples. On the other hand, an error of $O(εσ)$ can be information-theoretically attained with $O(d/ε^2)$ 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 $O(d^{2-c})$, where $c>0$ is an arbitrarily small constant, must incur an error of $Ω(\sqrtε σ)$.

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arXiv2018

Being Robust (in High Dimensions) Can Be Practical

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 high-dimensional 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 state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.

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arXiv2018

Fast and Sample Near-Optimal Algorithms for Learning Multidimensional Histograms

We study the problem of robustly learning multi-dimensional histograms. A $d$-dimensional function $h: D \rightarrow \mathbb{R}$ is called a $k$-histogram if there exists a partition of the domain $D \subseteq \mathbb{R}^d$ into $k$ axis-aligned rectangles such that $h$ is constant within each such rectangle. Let $f: D \rightarrow \mathbb{R}$ 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 $O(\mathrm{OPT}) + ε$ close to $f$, in $L_1$-distance. We give an algorithm for this learning problem that uses $n = \tilde{O}_d(k/ε^2)$ 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 near-linear time. Prior to our work, the time complexity of the $d=1$ case was well-understood, but significant gaps in our understanding remained even for $d=2$.

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arXiv2017

List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussians

We study the problem of list-decodable Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for these problems. {\bf List-Decodable Mean Estimation.} Fix any $d \in \mathbb{Z}_+$ and $0< α<1/2$. We design an algorithm with runtime $O (\mathrm{poly}(n/α)^{d})$ that outputs a list of $O(1/α)$ many candidate vectors such that with high probability one of the candidates is within $\ell_2$-distance $O(α^{-1/(2d)})$ from the true mean. The only previous algorithm for this problem achieved error $\tilde O(α^{-1/2})$ under second moment conditions. For $d = O(1/ε)$, our algorithm runs in polynomial time and achieves error $O(α^ε)$. We also give a Statistical Query lower bound suggesting that the complexity of our algorithm is qualitatively close to best possible. {\bf Learning Mixtures of Spherical Gaussians.} We give a learning algorithm for mixtures of spherical Gaussians that succeeds under significantly weaker separation assumptions compared to prior work. For the prototypical case of a uniform mixture of $k$ identity covariance Gaussians we obtain: For any $ε>0$, if the pairwise separation between the means is at least $Ω(k^ε+\sqrt{\log(1/δ)})$, our algorithm learns the unknown parameters within accuracy $δ$ with sample complexity and running time $\mathrm{poly} (n, 1/δ, (k/ε)^{1/ε})$. The previously best known polynomial time algorithm required separation at least $k^{1/4} \mathrm{polylog}(k/δ)$. Our main technical contribution is a new technique, using degree-$d$ multivariate polynomials, to remove outliers from high-dimensional datasets where the majority of the points are corrupted.

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arXiv2017

Robustly Learning a Gaussian: Getting Optimal Error, Efficiently

We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error $O(\varepsilon)$ 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 $1/ε$. When both the mean and covariance are unknown, the running time is polynomial in $d$ and quasipolynomial in $1/\varepsilon$. 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 one-dimensional setting can also be achieved by efficient algorithms in high-dimensional settings.

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arXiv2017

Sharp Bounds for Generalized Uniformity Testing

We study the problem of generalized uniformity testing \cite{BC17} of a discrete probability distribution: Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbfΩ$, we want to distinguish, with probability at least $2/3$, between the case that $p$ is uniform on some {\em subset} of $\mathbfΩ$ versus $ε$-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 information-theoretic lower bound. Specifically, we show that the sample complexity of generalized uniformity testing is $Θ\left(1/(ε^{4/3}\|p\|_3) + 1/(ε^{2} \|p\|_2) \right)$.

cs.DS↗

arXiv2017

Fourier-Based Testing for Families of Distributions

We study the general problem of testing whether an unknown distribution belongs to a specified family of distributions. More specifically, given a distribution family $\mathcal{P}$ and sample access to an unknown discrete distribution $\mathbf{P}$, we want to distinguish (with high probability) between the case that $\mathbf{P} \in \mathcal{P}$ and the case that $\mathbf{P}$ is $ε$-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 inference task depends on the underlying family $\mathcal{P}$. The gold standard in distribution property testing is to design sample-optimal and computationally efficient algorithms for this task. The main contribution of this work is a simple and general testing technique that is applicable to all distribution families whose Fourier spectrum satisfies a certain approximate sparsity property. To the best of our knowledge, ours is the first use of the Fourier transform in the context of distribution testing. We apply our Fourier-based framework to obtain near sample-optimal and computationally efficient testers for the following fundamental distribution families: Sums of Independent Integer Random Variables (SIIRVs), Poisson Multinomial Distributions (PMDs), and Discrete Log-Concave Distributions. For the first two, ours are the first non-trivial testers in the literature, vastly generalizing previous work on testing Poisson Binomial Distributions. For the third, our tester improves on prior work in both sample and time complexity.

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arXiv2017

Differentially Private Identity and Closeness Testing of Discrete Distributions

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 sample-efficient differentially private testers for these problems. Our theoretical results show that there exist private identity and closeness testers that are nearly as sample-efficient as their non-private 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 sublinear in the domain size of the underlying distributions.

cs.LG↗

arXiv2017

Learning Geometric Concepts with Nasty Noise

We study the efficient learnability of geometric concept classes - specifically, low-degree polynomial threshold functions (PTFs) and intersections of halfspaces - when a fraction of the data is adversarially corrupted. We give the first polynomial-time PAC learning algorithms for these concept classes with dimension-independent error guarantees in the presence of 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 well-studied 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 origin-centered halfspaces. Specifically, our robust learning algorithm for low-degree PTFs succeeds under a number of tame distributions -- including the Gaussian distribution and, more generally, any log-concave distribution with (approximately) known low-degree moments. For LTFs under the Gaussian distribution, we give a polynomial-time algorithm that achieves error $O(ε)$, where $ε$ is the noise rate. At the core of our PAC learning results is an efficient algorithm to approximate the low-degree Chow-parameters 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.

cs.LG↗

arXiv2017

Learning Multivariate Log-concave Distributions

We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave 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 $ε>0$, draws $\tilde{O}_d \left( (1/ε)^{(d+5)/2} \right)$ samples from an unknown target log-concave density on $\mathbb{R}^d$, and outputs a hypothesis that (with high probability) is $ε$-close to the target, in total variation distance. Our upper bound on the sample complexity comes close to the known lower bound of $Ω_d \left( (1/ε)^{(d+1)/2} \right)$ for this problem.

cs.LG↗

arXiv2017

Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures

We describe a general technique that yields the first {\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown Gaussian distribution. For each of these problems, we show a {\em super-polynomial gap} between the (information-theoretic) sample complexity and the computational complexity of {\em any} Statistical Query algorithm for the problem. Our SQ lower bound for Problem (1) is qualitatively matched by known learning algorithms for GMMs. Our lower bound for Problem (2) implies that the accuracy of the robust learning algorithm in~\cite{DiakonikolasKKLMS16} is essentially best possible among all polynomial-time SQ algorithms. Our SQ lower bounds are attained via a unified moment-matching technique that is useful in other contexts and may be of broader interest. Our technique yields nearly-tight lower bounds for a number of related unsupervised estimation problems. Specifically, for the problems of (3) robust covariance estimation in spectral norm, and (4) robust sparse mean estimation, we establish a quadratic {\em statistical--computational tradeoff} for SQ algorithms, matching known upper bounds. Finally, our technique can be used to obtain tight sample complexity lower bounds for high-dimensional {\em testing} problems. Specifically, for the classical problem of robustly {\em testing} an unknown mean (known covariance) Gaussian, our technique implies an information-theoretic sample lower bound that scales {\em linearly} in the dimension. Our sample lower bound matches the sample complexity of the corresponding robust {\em learning} problem and separates the sample complexity of robust testing from standard (non-robust) testing.

cs.LG↗

arXiv2017

Near-Optimal Closeness Testing of Discrete Histogram Distributions

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 $\|p-q\|_1 \geq ε$. The main contribution of this paper is a new algorithm for this testing problem and a nearly matching information-theoretic 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 setting and yields improved sample upper bounds for testing closeness of other structured distributions as well.

cs.DS↗

arXiv2016

Collision-based Testers are Optimal for Uniformity and Closeness

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 sample-optimal estimators are known for them~\cite{Paninski:08, CDVV14, VV14, DKN:15}. In this work, we show that the original collision-based testers proposed for these problems ~\cite{GRdist:00, BFR+:00} are sample-optimal, 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 $ε$. Our main contribution is a new tight analysis establishing that these collision-based testers are information-theoretically optimal, up to constant factors, both in the dependence on $n$ and in the dependence on $ε$.

cs.DS↗

arXiv2016

Near-Optimal Disjoint-Path Facility Location Through Set Cover by Pairs

In this paper we consider two special cases of the "cover-by-pairs" optimization problem that arise when we need to place facilities so that each customer is served by two facilities that reach it by disjoint shortest paths. These problems arise in a network traffic monitoring scheme proposed by Breslau et al. and have potential applications to content distribution. The "set-disjoint" variant applies to networks that use the OSPF routing protocol, and the "path-disjoint" variant applies when MPLS routing is enabled, making better solutions possible at the cost of greater operational expense. Although we can prove that no polynomial-time algorithm can guarantee good solutions for either version, we are able to provide heuristics that do very well in practice on instances with real-world network structure. Fast implementations of the heuristics, made possible by exploiting mathematical observations about the relationship between the network instances and the corresponding instances of the cover-by-pairs problem, allow us to perform an extensive experimental evaluation of the heuristics and what the solutions they produce tell us about the effectiveness of the proposed monitoring scheme. For the set-disjoint variant, we validate our claim of near-optimality via a new lower-bounding integer programming formulation. Although computing this lower bound requires solving the NP-hard Hitting Set problem and can underestimate the optimal value by a linear factor in the worst case, it can be computed quickly by CPLEX, and it equals the optimal solution value for all the instances in our extensive testbed.

cs.DS↗

arXiv2016

Playing Anonymous Games using Simple Strategies

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 $δ>0$, an $O(1/n^{1-δ})$-approximate Nash equilibrium can be computed in polynomial time. Complementing this positive result, we show that if there exists any constant $δ>0$ such that an $O(1/n^{1+δ})$-approximate equilibrium can be computed in polynomial time, then there is a fully polynomial-time 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 $1-δ$, for some small $δ$, and plays uniformly at random with probability $δ$. 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.

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