arXiv subjects

Ilias Diakonikolas

Publications and source records attributed to Ilias Diakonikolas.

At least 127 records · Page 7Linked to original sources

arXiv2020

Learning Halfspaces with Massart Noise Under Structured Distributions

We study the problem of learning halfspaces with Massart noise in the distribution-specific PAC model. We give the first computationally efficient algorithm for this problem with respect to a broad family of distributions, including log-concave distributions. This resolves an open question posed in a number of prior works. Our approach is extremely simple: We identify a smooth {\em non-convex} 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.

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arXiv2020

Testing Bayesian Networks

This work initiates a systematic investigation of testing high-dimensional structured distributions by focusing on testing 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 non-descendant 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 non-trivial efficient testing algorithms for these problems and corresponding information-theoretic lower bounds. For a wide range of parameter settings, our testing algorithms have sample complexity sublinear in the dimension and are sample-optimal, up to constant factors.

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arXiv2019

Distribution-Independent PAC Learning of Halfspaces with Massart Noise

We study the problem of {\em distribution-independent} PAC learning of halfspaces in the presence of Massart noise. Specifically, we are given a set of labeled examples $(\mathbf{x}, y)$ drawn from a distribution $\mathcal{D}$ on $\mathbb{R}^{d+1}$ such that the marginal distribution on the unlabeled points $\mathbf{x}$ is arbitrary and the labels $y$ are generated by an unknown halfspace corrupted with Massart noise at noise rate $η<1/2$. The goal is to find a hypothesis $h$ that minimizes the misclassification error $\mathbf{Pr}_{(\mathbf{x}, y) \sim \mathcal{D}} \left[ h(\mathbf{x}) \neq y \right]$. We give a $\mathrm{poly}\left(d, 1/ε\right)$ time algorithm for this problem with misclassification error $η+ε$. We also provide evidence that improving on the error guarantee of our algorithm might be computationally hard. Prior to our work, no efficient weak (distribution-independent) 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.

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arXiv2019

Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering

We study high-dimensional 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 near-optimal 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.

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arXiv2019

Recent Advances in Algorithmic High-Dimensional Robust Statistics

Learning in the presence of outliers is a fundamental problem in statistics. Until recently, all known efficient unsupervised learning algorithms were very sensitive to outliers in high dimensions. In particular, even for the task of robust mean estimation under natural distributional assumptions, no efficient algorithm was known. Recent work in theoretical computer science gave the first efficient robust estimators for a number of fundamental statistical tasks, including mean and covariance estimation. Since then, there has been a flurry of research activity on algorithmic high-dimensional robust estimation in a range of settings. In this survey article, we introduce the core ideas and algorithmic techniques in the emerging area of algorithmic high-dimensional robust statistics with a focus on robust mean estimation. We also provide an overview of the approaches that have led to computationally efficient robust estimators for a range of broader statistical tasks and discuss new directions and opportunities for future work.

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arXiv2019

Equipping Experts/Bandits with Long-term Memory

We propose the first reduction-based approach to obtaining long-term 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 $\mathcal{O}(\sqrt{T(S\ln T + n \ln K)})$ compared to any sequence of experts with $S-1$ 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 multi-armed bandit setting and show both negative and positive results for long-term memory guarantees. As a side result, our lower bound also implies that sparse losses do not help improve the worst-case regret for contextual bandits, a sharp contrast with the non-contextual case.

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arXiv2019

Nearly Tight Bounds for Robust Proper Learning of Halfspaces with a Margin

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 $α\cdot \mathrm{OPT}_γ + ε$, where $\mathrm{OPT}_γ$ is the optimal $γ$-margin error rate and $α\geq 1$ is the approximation ratio. We give learning algorithms and computational hardness results for this problem, for all values of the approximation ratio $α\geq 1$, that are nearly-matching for a range of parameters. Specifically, for the natural setting that $α$ is any constant bigger than one, we provide an essentially tight complexity characterization. On the positive side, we give an $α= 1.01$-approximate proper learner that uses $O(1/(ε^2γ^2))$ samples (which is optimal) and runs in time $\mathrm{poly}(d/ε) \cdot 2^{\tilde{O}(1/γ^2)}$. On the negative side, we show that {\em any} constant factor approximate proper learner has runtime $\mathrm{poly}(d/ε) \cdot 2^{(1/γ)^{2-o(1)}}$, assuming the Exponential Time Hypothesis.

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arXiv2019

A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential Families

We consider the problem of computing the maximum likelihood multivariate log-concave distribution for a set of points. Specifically, we present an algorithm which, given $n$ points in $\mathbb{R}^d$ and an accuracy parameter $ε>0$, runs in time $poly(n,d,1/ε),$ and returns a log-concave distribution which, with high probability, has the property that the likelihood of the $n$ points under the returned distribution is at most an additive $ε$ less than the maximum likelihood that could be achieved via any log-concave distribution. This is the first computationally efficient (polynomial time) algorithm for this fundamental and practically important task. Our algorithm rests on a novel connection with exponential families: the maximum likelihood log-concave distribution belongs to a class of structured distributions which, while not an exponential family, "locally" possesses key properties of exponential families. This connection then allows the problem of computing the log-concave maximum likelihood distribution to be formulated as a convex optimization problem, and solved via an approximate first-order method. Efficiently approximating the (sub) gradients of the objective function of this optimization problem is quite delicate, and is the main technical challenge in this work.

cs.DS↗

arXiv2019

Faster Algorithms for High-Dimensional Robust Covariance Estimation

We study the problem of estimating the covariance matrix of a high-dimensional 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 near-optimal 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 $N = \tildeΩ(d^2/ε^2)$ samples from a $d$-dimensional Gaussian distribution, an $ε$-fraction of which may be arbitrarily corrupted, our algorithm runs in time $\tilde{O}(d^{3.26})/\mathrm{poly}(ε)$ and approximates the unknown covariance matrix to optimal error up to a logarithmic factor. Previous robust algorithms with comparable error guarantees all have runtimes $\tildeΩ(d^{2 ω})$ when $ε= Ω(1)$, where $ω$ is the exponent of matrix multiplication. We also provide evidence that improving the running time of our algorithm may require new algorithmic techniques.

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arXiv2019

Communication and Memory Efficient Testing of Discrete Distributions

We study distribution testing with communication and memory constraints in the following computational models: (1) The {\em one-pass 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 nearly-tight lower bounds on (1) the sample complexity of any one-pass streaming tester for uniformity, subject to the memory constraint, and (2) the communication cost of any uniformity testing protocol, in a restricted `one-pass' model of communication.

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arXiv2019

Sever: A Robust Meta-Algorithm for Stochastic Optimization

In high dimensions, most machine learning methods are brittle to even a small fraction of structured outliers. To address this, we introduce a new meta-algorithm that can take in a 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 scalable -- beyond 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 $1.42$ mean-squared error test error, compared to $1.51$-$2.33$ for the baselines, and $1.23$ error on the uncorrupted dataset.

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arXiv2019

Robust Estimators in High Dimensions without the Computational Intractability

We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an $\varepsilon$-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 high-dimensional agnostic distribution learning even possible, algorithmically? In this work, we obtain the first computationally efficient algorithms with dimension-independent error guarantees for agnostically learning several fundamental classes of high-dimensional 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 nearly-linearly 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.

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arXiv2019

Testing Identity of Multidimensional Histograms

We investigate the problem of identity testing for multidimensional histogram distributions. A distribution $p: D \rightarrow \mathbb{R}_+$, where $D \subseteq \mathbb{R}^d$, is called a $k$-histogram if there exists a partition of the domain into $k$ axis-aligned 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 sub-learning} sample complexity in any fixed dimension and a nearly-matching sample complexity lower bound. In more detail, let $q$ be an unknown $d$-dimensional $k$-histogram distribution in fixed dimension $d$, and $p$ be an explicitly given $d$-dimensional $k$-histogram. We want to correctly distinguish, with probability at least $2/3$, between the case that $p = q$ versus $\|p-q\|_1 \geq ε$. We design an algorithm for this hypothesis testing problem with sample complexity $O((\sqrt{k}/ε^2) 2^{d/2} \log^{2.5 d}(k/ε))$ that runs in sample-polynomial time. 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 $k = 2^{Ω(d)}$, we show a sample complexity lower bound of $(\sqrt{k}/ε^2) \cdot Ω(\log(k)/d)^{d-1}$ when $d\geq 2$. That is, for any fixed dimension $d$, our upper and lower bounds are nearly matching. Prior to our work, the sample complexity of the $d=1$ case was well-understood, but no algorithm with sub-learning 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.

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arXiv2019

Optimal Identity Testing with High Probability

We study the problem of testing identity against a given distribution 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 $0< ε, δ< 1$, we wish to distinguish, {\em with probability at least $1-δ$}, whether the distributions are identical versus $\varepsilon$-far in total variation distance. Most prior work focused on the case that $δ= Ω(1)$, for which the sample complexity of identity testing is known to be $Θ(\sqrt{n}/ε^2)$. Given such an algorithm, one can achieve arbitrarily small values of $δ$ via black-box amplification, which multiplies the required number of samples by $Θ(\log(1/δ))$. We show that black-box amplification is suboptimal for any $δ= o(1)$, 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 \[ Θ\left( \frac{1}{ε^2}\left(\sqrt{n \log(1/δ)} + \log(1/δ) \right)\right) \] for any $n, \varepsilon$, and $δ$. 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 $d_{\mathrm TV}(p, U_n) \geq \varepsilon$, we simply threshold $d_{\mathrm TV}(\widehat{p}, U_n)$, where $\widehat{p}$ is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant $δ$ case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of $\varepsilon$ and $δ$.

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arXiv2018

On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games

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.

cs.GT↗

arXiv2018

A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities

We study the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave 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 $\mathbb{R}^d$ and an accuracy parameter $ε>0$, it runs in time $\text{poly}(n, d, 1/ε)$, and outputs a log-concave density that with high probability maximizes the log-likelihood up to an additive $ε$. 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 log-concave densities and leverage classical algorithmic tools on volume approximation of convex bodies and uniform sampling from convex sets.

cs.DS↗

arXiv2018

Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities

We study the problem of learning multivariate log-concave 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 log-concave 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 $ε>0$, given $\tilde{O}_d((1/ε)^{(d+3)/2})$ samples drawn from an unknown log-concave density $f_0$ on $\mathbb{R}^d$, the MLE outputs a hypothesis $h$ that with high probability is $ε$-close to $f_0$, in squared Hellinger loss. A sample complexity lower bound of $Ω_d((1/ε)^{(d+1)/2})$ was previously known for any learning algorithm that achieves this guarantee. We thus establish that the sample complexity of the log-concave MLE is near-optimal, up to an $\tilde{O}(1/ε)$ factor.

math.ST↗

arXiv2018

High-Dimensional Robust Mean Estimation in Nearly-Linear Time

We study the fundamental problem of high-dimensional 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 dimension-independent error guarantees for several families of structured distributions. In this work, we give the first nearly-linear time algorithms for high-dimensional robust mean estimation. Specifically, we focus on distributions with (i) known covariance and sub-gaussian tails, and (ii) unknown bounded covariance. Given $N$ samples on $\mathbb{R}^d$, an $ε$-fraction of which may be arbitrarily corrupted, our algorithms run in time $\tilde{O}(Nd) / \mathrm{poly}(ε)$ and approximate the true mean within the information-theoretically optimal error, up to constant factors. Previous robust algorithms with comparable error guarantees have running times $\tildeΩ(N d^2)$, for $ε= Ω(1)$. Our algorithms rely on a natural family of SDPs parameterized by our current guess $ν$ for the unknown mean $μ^\star$. We give a win-win analysis establishing the following: either a near-optimal solution to the primal SDP yields a good candidate for $μ^\star$ -- independent of our current guess $ν$ -- or the dual SDP yields a new guess $ν'$ whose distance from $μ^\star$ is smaller by a constant factor. We exploit the special structure of the corresponding SDPs to show that they are approximately solvable in nearly-linear time. Our approach is quite general, and we believe it can also be applied to obtain nearly-linear time algorithms for other high-dimensional robust learning problems.

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