arXiv subjects

Ilias Diakonikolas

Publications and source records attributed to Ilias Diakonikolas.

At least 109 records · Page 6Linked to original sources

arXiv2021

Outlier-Robust Learning of Ising Models Under Dobrushin's Condition

We study the problem of learning Ising models satisfying Dobrushin's condition in the outlier-robust setting where a constant fraction of the samples are adversarially corrupted. Our main result is to provide the first computationally efficient robust learning algorithm for this problem with near-optimal error guarantees. Our algorithm can be seen as a special case of an algorithm for robustly learning a distribution from a general exponential family. To prove its correctness for Ising models, we establish new anti-concentration results for degree-$2$ polynomials of Ising models that may be of independent interest.

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arXiv2020

The Sample Complexity of Robust Covariance Testing

We study the problem of testing the covariance matrix of a high-dimensional Gaussian in a robust setting, where the input distribution has been corrupted in Huber's contamination model. Specifically, we are given i.i.d. samples from a distribution of the form $Z = (1-ε) X + εB$, where $X$ is a zero-mean and unknown covariance Gaussian $\mathcal{N}(0, Σ)$, $B$ is a fixed but unknown noise distribution, and $ε>0$ is an arbitrarily small constant representing the proportion of contamination. We want to distinguish between the cases that $Σ$ is the identity matrix versus $γ$-far from the identity in Frobenius norm. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses $O(d)$ samples. Moreover, this sample upper bound was shown to be best possible, within constant factors. Our main result is that the sample complexity of covariance testing dramatically increases in the contaminated setting. In particular, we prove a sample complexity lower bound of $Ω(d^2)$ for $ε$ an arbitrarily small constant and $γ= 1/2$. This lower bound is best possible, as $O(d^2)$ samples suffice to even robustly {\em learn} the covariance. The conceptual implication of our result is that, for the natural setting we consider, robust hypothesis testing is at least as hard as robust estimation.

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arXiv2020

Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models

Let $V$ be any vector space of multivariate degree-$d$ homogeneous polynomials with co-dimension at most $k$, and $S$ be the set of points where all polynomials in $V$ {\em nearly} vanish. We establish a qualitatively optimal upper bound on the size of $ε$-covers for $S$, in the $\ell_2$-norm. Roughly speaking, we show that there exists an $ε$-cover for $S$ of cardinality $M = (k/ε)^{O_d(k^{1/d})}$. Our result is constructive yielding an algorithm to compute such an $ε$-cover that runs in time $\mathrm{poly}(M)$. Building on our structural result, we obtain significantly improved learning algorithms for several fundamental high-dimensional probabilistic models with hidden variables. These include density and parameter estimation for $k$-mixtures of spherical Gaussians (with known common covariance), PAC learning one-hidden-layer ReLU networks with $k$ hidden units (under the Gaussian distribution), density and parameter estimation for $k$-mixtures of linear regressions (with Gaussian covariates), and parameter estimation for $k$-mixtures of hyperplanes. Our algorithms run in time {\em quasi-polynomial} in the parameter $k$. Previous algorithms for these problems had running times exponential in $k^{Ω(1)}$. At a high-level our algorithms for all these learning problems work as follows: By computing the low-degree moments of the hidden parameters, we are able to find a vector space of polynomials that nearly vanish on the unknown parameters. Our structural result allows us to compute a quasi-polynomial sized cover for the set of hidden parameters, which we exploit in our learning algorithms.

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arXiv2020

List-Decodable Mean Estimation in Nearly-PCA Time

Traditionally, robust statistics has focused on designing estimators tolerant to a minority of contaminated data. Robust list-decodable learning focuses on the more challenging regime where only a minority $\frac 1 k$ fraction of the dataset is drawn from the distribution of interest, and no assumptions are made on the remaining data. We study the fundamental task of list-decodable mean estimation in high dimensions. Our main result is a new list-decodable mean estimation algorithm for bounded covariance distributions with optimal sample complexity and error rate, running in nearly-PCA time. Assuming the ground truth distribution on $\mathbb{R}^d$ has bounded covariance, our algorithm outputs a list of $O(k)$ candidate means, one of which is within distance $O(\sqrt{k})$ from the truth. Our algorithm runs in time $\widetilde{O}(ndk)$ for all $k = O(\sqrt{d}) \cup Ω(d)$, where $n$ is the size of the dataset. We also show that a variant of our algorithm has runtime $\widetilde{O}(ndk)$ for all $k$, at the expense of an $O(\sqrt{\log k})$ factor in the recovery guarantee. This runtime matches up to logarithmic factors the cost of performing a single $k$-PCA on the data, which is a natural bottleneck of known algorithms for (very) special cases of our problem, such as clustering well-separated mixtures. Prior to our work, the fastest list-decodable mean estimation algorithms had runtimes $\widetilde{O}(n^2 d k^2)$ and $\widetilde{O}(nd k^{\ge 6})$. Our approach builds on a novel soft downweighting method, $\mathsf{SIFT}$, which is arguably the simplest known polynomial-time mean estimation technique in the list-decodable learning setting. To develop our fast algorithms, we boost the computational cost of $\mathsf{SIFT}$ via a careful "win-win-win" analysis of an approximate Ky Fan matrix multiplicative weights procedure we develop, which we believe may be of independent interest.

cs.DS↗

arXiv2020

A Polynomial Time Algorithm for Learning Halfspaces with Tsybakov Noise

We study the problem of PAC learning homogeneous halfspaces in the presence of Tsybakov noise. In the Tsybakov noise model, the label of every sample is independently flipped with an adversarially controlled probability that can be arbitrarily close to $1/2$ for a fraction of the samples. {\em We give the first polynomial-time algorithm for this fundamental learning problem.} Our algorithm learns the true halfspace within any desired accuracy $ε$ and succeeds under a broad family of well-behaved distributions including log-concave distributions. Prior to our work, the only previous algorithm for this problem required quasi-polynomial runtime in $1/ε$. Our algorithm employs a recently developed reduction \cite{DKTZ20b} from learning to certifying the non-optimality of a candidate halfspace. This prior work developed a quasi-polynomial time certificate algorithm based on polynomial regression. {\em The main technical contribution of the current paper is the first polynomial-time certificate algorithm.} Starting from a non-trivial warm-start, our algorithm performs a novel "win-win" iterative process which, at each step, either finds a valid certificate or improves the angle between the current halfspace and the true one. Our warm-start algorithm for isotropic log-concave distributions involves a number of analytic tools that may be of broader interest. These include a new efficient method for reweighting the distribution in order to recenter it and a novel characterization of the spectrum of the degree-$2$ Chow parameters.

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arXiv2020

Approximation Schemes for ReLU Regression

We consider the fundamental problem of ReLU regression, where the goal is to output the best fitting ReLU with respect to square loss given access to draws from some unknown distribution. We give the first efficient, constant-factor approximation algorithm for this problem assuming the underlying distribution satisfies some weak concentration and anti-concentration conditions (and includes, for example, all log-concave distributions). This solves the main open problem of Goel et al., who proved hardness results for any exact algorithm for ReLU regression (up to an additive $ε$). Using more sophisticated techniques, we can improve our results and obtain a polynomial-time approximation scheme for any subgaussian distribution. Given the aforementioned hardness results, these guarantees can not be substantially improved. Our main insight is a new characterization of surrogate losses for nonconvex activations. While prior work had established the existence of convex surrogates for monotone activations, we show that properties of the underlying distribution actually induce strong convexity for the loss, allowing us to relate the global minimum to the activation's Chow parameters.

cs.LG↗

arXiv2020

Optimal Testing of Discrete Distributions with High Probability

We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property $\mathcal{P}$, and parameters $0< ε, δ<1$, we want to distinguish {\em with probability at least $1-δ$} whether these distributions satisfy $\mathcal{P}$ or are $ε$-far from $\mathcal{P}$ in total variation distance. Most prior work in distribution testing studied the constant confidence case (corresponding to $δ= Ω(1)$), and provided sample-optimal testers for a range of properties. While one can always boost the confidence probability of any such tester by black-box amplification, this generic boosting method typically leads to sub-optimal sample bounds. Here we study the following broad question: For a given property $\mathcal{P}$, can we {\em characterize} the sample complexity of testing $\mathcal{P}$ as a function of all relevant problem parameters, including the error probability $δ$? Prior to this work, uniformity testing was the only statistical task whose sample complexity had been characterized in this setting. As our main results, we provide the first algorithms for closeness and independence testing that are sample-optimal, within constant factors, as a function of all relevant parameters. We also show matching information-theoretic lower bounds on the sample complexity of these problems. Our techniques naturally extend to give optimal testers for related problems. To illustrate the generality of our methods, we give optimal algorithms for testing collections of distributions and testing closeness with unequal sized samples.

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arXiv2020

Rapid Approximate Aggregation with Distribution-Sensitive Interval Guarantees

Aggregating data is fundamental to data analytics, data exploration, and OLAP. Approximate query processing (AQP) techniques are often used to accelerate computation of aggregates using samples, for which confidence intervals (CIs) are widely used to quantify the associated error. CIs used in practice fall into two categories: techniques that are tight but not correct, i.e., they yield tight intervals but only offer asymptotic guarantees, making them unreliable, or techniques that are correct but not tight, i.e., they offer rigorous guarantees, but are overly conservative, leading to confidence intervals that are too loose to be useful. In this paper, we develop a CI technique that is both correct and tighter than traditional approaches. Starting from conservative CIs, we identify two issues they often face: pessimistic mass allocation (PMA) and phantom outlier sensitivity (PHOS). By developing a novel range-trimming technique for eliminating PHOS and pairing it with known CI techniques without PMA, we develop a technique for computing CIs with strong guarantees that requires fewer samples for the same width. We implement our techniques underneath a sampling-optimized in-memory column store and show how to accelerate queries involving aggregates on a real dataset with speedups of up to 124x over traditional AQP-with-guarantees and more than 1000x over exact methods.

cs.DB↗

arXiv2020

The Complexity of Adversarially Robust Proper Learning of Halfspaces with Agnostic Noise

We study the computational complexity of adversarially robust proper learning of halfspaces in the distribution-independent agnostic PAC model, with a focus on $L_p$ perturbations. We give a computationally efficient learning algorithm and a nearly matching computational hardness result for this problem. An interesting implication of our findings is that the $L_{\infty}$ perturbations case is provably computationally harder than the case $2 \leq p < \infty$.

cs.LG↗

arXiv2020

Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals

We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \mathbb{R}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with square loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the square loss of the best-fitting ReLU. We prove Statistical Query (SQ) lower bounds of $d^{\mathrm{poly}(1/ε)}$ for both of these problems. Our SQ lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.

cs.LG↗

arXiv2020

Algorithms and SQ Lower Bounds for PAC Learning One-Hidden-Layer ReLU Networks

We study the problem of PAC learning one-hidden-layer ReLU networks with $k$ hidden units on $\mathbb{R}^d$ under Gaussian marginals in the presence of additive label noise. For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$. Previously, no polynomial time algorithm was known, even for $k=3$. This answers an open question posed by~\cite{Kliv17}. Importantly, our algorithm does not require any assumptions about the rank of the weight matrix and its complexity is independent of its condition number. On the negative side, for the more general task of PAC learning one-hidden-layer ReLU networks with arbitrary real coefficients, we prove a Statistical Query lower bound of $d^{Ω(k)}$. Thus, we provide a separation between the two classes in terms of efficient learnability. Our upper and lower bounds are general, extending to broader families of activation functions.

cs.LG↗

arXiv2020

List-Decodable Mean Estimation via Iterative Multi-Filtering

We study the problem of {\em list-decodable mean estimation} for bounded covariance distributions. Specifically, we are given a set $T$ of points in $\mathbb{R}^d$ with the promise that an unknown $α$-fraction of points in $T$, where $0< α< 1/2$, are drawn from an unknown mean and bounded covariance distribution $D$, and no assumptions are made on the remaining points. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the mean of $D$. We give the first practically viable estimator for this problem. In more detail, our algorithm is sample and computationally efficient, and achieves information-theoretically near-optimal error. While the only prior algorithm for this setting inherently relied on the ellipsoid method, our algorithm is iterative and only uses spectral techniques. Our main technical innovation is the design of a soft outlier removal procedure for high-dimensional heavy-tailed datasets with a majority of outliers.

cs.LG↗

arXiv2020

Learning Halfspaces with Tsybakov Noise

We study the efficient PAC learnability of halfspaces in the presence of Tsybakov noise. In the Tsybakov noise model, each label is independently flipped with some probability which is controlled by an adversary. This noise model significantly generalizes the Massart noise model, by allowing the flipping probabilities to be arbitrarily close to $1/2$ for a fraction of the samples. Our main result is the first non-trivial PAC learning algorithm for this problem under a broad family of structured distributions -- satisfying certain concentration and (anti-)anti-concentration properties -- including log-concave distributions. Specifically, we given an algorithm that achieves misclassification error $ε$ with respect to the true halfspace, with quasi-polynomial runtime dependence in $1/\epsilin$. The only previous upper bound for this problem -- even for the special case of log-concave distributions -- was doubly exponential in $1/ε$ (and follows via the naive reduction to agnostic learning). Our approach relies on a novel computationally efficient procedure to certify whether a candidate solution is near-optimal, based on semi-definite programming. We use this certificate procedure as a black-box and turn it into an efficient learning algorithm by searching over the space of halfspaces via online convex optimization.

cs.LG↗

arXiv2020

Non-Convex SGD Learns Halfspaces with Adversarial Label Noise

We study the problem of agnostically learning homogeneous halfspaces in the distribution-specific PAC model. For a broad family of structured distributions, including log-concave distributions, we show that non-convex SGD efficiently converges to a solution with misclassification error $O(\opt)+\eps$, where $\opt$ is the misclassification error of the best-fitting halfspace. In sharp contrast, we show that optimizing any convex surrogate inherently leads to misclassification error of $ω(\opt)$, even under Gaussian marginals.

cs.LG↗

arXiv2020

Efficiently Learning Adversarially Robust Halfspaces with Noise

We study the problem of learning adversarially robust halfspaces in the distribution-independent setting. In the realizable setting, we provide necessary and sufficient conditions on the adversarial perturbation sets under which halfspaces are efficiently robustly learnable. In the presence of random label noise, we give a simple computationally efficient algorithm for this problem with respect to any $\ell_p$-perturbation.

cs.LG↗

arXiv2020

Robustly Learning any Clusterable Mixture of Gaussians

We study the efficient learnability of high-dimensional Gaussian mixtures in the outlier-robust 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 $ε$-corrupted $k$-mixture within information theoretically near-optimal error of $\tilde{O}(ε)$, under the assumption that the overlap between any pair of components $P_i, P_j$ (i.e., the quantity $1-TV(P_i, P_j)$) is bounded by $\mathrm{poly}(ε)$. 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 Sum-of-Squares based 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 constant-degree Sum of Squares proof system. The key ingredients of this proof are a novel use of SoS-certifiable anti-concentration and a new characterization of pairs of Gaussians with small (dimension-independent) overlap in terms of their parameter distance.

cs.DS↗

arXiv2020

High-Dimensional Robust Mean Estimation via Gradient Descent

We study the problem of high-dimensional robust mean estimation in the presence of a constant fraction of adversarial outliers. A recent line of work has provided sophisticated polynomial-time algorithms for this problem with dimension-independent error guarantees for a range of natural distribution families. In this work, we show that a natural non-convex 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 non-convex objective gives a near-optimal solution to the underlying robust estimation task. Our work establishes an intriguing connection between algorithmic high-dimensional robust statistics and non-convex optimization, which may have broader applications to other robust estimation tasks.

cs.LG↗

arXiv2020

Efficient Algorithms for Multidimensional Segmented Regression

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 mean-squared 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 state-of-the-art heuristics. Code of our implementation is available at \url{https://github.com/avoloshinov/multidimensional-segmented-regression}.

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