arXiv subjects

Ilias Diakonikolas

Publications and source records attributed to Ilias Diakonikolas.

At least 55 records · Page 3Linked to original sources

arXiv2024

SoS Certifiability of Subgaussian Distributions and its Algorithmic Applications

We prove that there is a universal constant $C>0$ so that for every $d \in \mathbb N$, every centered subgaussian distribution $\mathcal D$ on $\mathbb R^d$, and every even $p \in \mathbb N$, the $d$-variate polynomial $(Cp)^{p/2} \cdot \|v\|_{2}^p - \mathbb E_{X \sim \mathcal D} \langle v,X\rangle^p$ is a sum of square polynomials. This establishes that every subgaussian distribution is \emph{SoS-certifiably subgaussian} -- a condition that yields efficient learning algorithms for a wide variety of high-dimensional statistical tasks. As a direct corollary, we obtain computationally efficient algorithms with near-optimal guarantees for the following tasks, when given samples from an arbitrary subgaussian distribution: robust mean estimation, list-decodable mean estimation, clustering mean-separated mixture models, robust covariance-aware mean estimation, robust covariance estimation, and robust linear regression. Our proof makes essential use of Talagrand's generic chaining/majorizing measures theorem.

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arXiv2024

Sum-of-squares lower bounds for Non-Gaussian Component Analysis

Non-Gaussian Component Analysis (NGCA) is the statistical task of finding a non-Gaussian direction in a high-dimensional dataset. Specifically, given i.i.d.\ samples from a distribution $P^A_{v}$ on $\mathbb{R}^n$ that behaves like a known distribution $A$ in a hidden direction $v$ and like a standard Gaussian in the orthogonal complement, the goal is to approximate the hidden direction. The standard formulation posits that the first $k-1$ moments of $A$ match those of the standard Gaussian and the $k$-th moment differs. Under mild assumptions, this problem has sample complexity $O(n)$. On the other hand, all known efficient algorithms require $Ω(n^{k/2})$ samples. Prior work developed sharp Statistical Query and low-degree testing lower bounds suggesting an information-computation tradeoff for this problem. Here we study the complexity of NGCA in the Sum-of-Squares (SoS) framework. Our main contribution is the first super-constant degree SoS lower bound for NGCA. Specifically, we show that if the non-Gaussian distribution $A$ matches the first $(k-1)$ moments of $\mathcal{N}(0, 1)$ and satisfies other mild conditions, then with fewer than $n^{(1 - \varepsilon)k/2}$ many samples from the normal distribution, with high probability, degree $(\log n)^{{1\over 2}-o_n(1)}$ SoS fails to refute the existence of such a direction $v$. Our result significantly strengthens prior work by establishing a super-polynomial information-computation tradeoff against a broader family of algorithms. As corollaries, we obtain SoS lower bounds for several problems in robust statistics and the learning of mixture models. Our SoS lower bound proof introduces a novel technique, that we believe may be of broader interest, and a number of refinements over existing methods.

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arXiv2024

Efficient Testable Learning of General Halfspaces with Adversarial Label Noise

We study the task of testable learning of general -- not necessarily homogeneous -- halfspaces with adversarial label noise with respect to the Gaussian distribution. In the testable learning framework, the goal is to develop a tester-learner such that if the data passes the tester, then one can trust the output of the robust learner on the data.Our main result is the first polynomial time tester-learner for general halfspaces that achieves dimension-independent misclassification error. At the heart of our approach is a new methodology to reduce testable learning of general halfspaces to testable learning of nearly homogeneous halfspaces that may be of broader interest.

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arXiv2024

First Order Stochastic Optimization with Oblivious Noise

We initiate the study of stochastic optimization with oblivious noise, broadly generalizing the standard heavy-tailed noise setup. In our setting, in addition to random observation noise, the stochastic gradient may be subject to independent oblivious noise, which may not have bounded moments and is not necessarily centered. Specifically, we assume access to a noisy oracle for the stochastic gradient of $f$ at $x$, which returns a vector $\nabla f(γ, x) + ξ$, where $γ$ is the bounded variance observation noise and $ξ$ is the oblivious noise that is independent of $γ$ and $x$. The only assumption we make on the oblivious noise $ξ$ is that $\mathbf{Pr}[ξ= 0] \ge α$ for some $α\in (0, 1)$. In this setting, it is not information-theoretically possible to recover a single solution close to the target when the fraction of inliers $α$ is less than $1/2$. Our main result is an efficient list-decodable learner that recovers a small list of candidates, at least one of which is close to the true solution. On the other hand, if $α= 1-ε$, where $0< ε< 1/2$ is sufficiently small constant, the algorithm recovers a single solution. Along the way, we develop a rejection-sampling-based algorithm to perform noisy location estimation, which may be of independent interest.

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arXiv2024

Robust Sparse Mean Estimation via Sum of Squares

We study the problem of high-dimensional sparse mean estimation in the presence of an $ε$-fraction of adversarial outliers. Prior work obtained sample and computationally efficient algorithms for this task for identity-covariance subgaussian distributions. In this work, we develop the first efficient algorithms for robust sparse mean estimation without a priori knowledge of the covariance. For distributions on $\mathbb R^d$ with "certifiably bounded" $t$-th moments and sufficiently light tails, our algorithm achieves error of $O(ε^{1-1/t})$ with sample complexity $m = (k\log(d))^{O(t)}/ε^{2-2/t}$. For the special case of the Gaussian distribution, our algorithm achieves near-optimal error of $\tilde O(ε)$ with sample complexity $m = O(k^4 \mathrm{polylog}(d))/ε^2$. Our algorithms follow the Sum-of-Squares based, proofs to algorithms approach. We complement our upper bounds with Statistical Query and low-degree polynomial testing lower bounds, providing evidence that the sample-time-error tradeoffs achieved by our algorithms are qualitatively the best possible.

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arXiv2024

List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering

We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $α\in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor αm \rfloor$ of which are i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $μ$. No assumptions are made on the remaining points, which form the majority of the dataset. The goal is to return a small list of candidates containing a vector $\widehat μ$ such that $\| \widehat μ- μ\|_2$ is small. Prior work had studied the problem of list-decodable mean estimation in the dense setting. In this work, we develop a novel, conceptually simpler technique for list-decodable mean estimation. As the main application of our approach, we provide the first sample and computationally efficient algorithm for list-decodable sparse mean estimation. In particular, for distributions with "certifiably bounded" $t$-th moments in $k$-sparse directions and sufficiently light tails, our algorithm achieves error of $(1/α)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/α$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of $Θ(\sqrt{\log(1/α)})$ with quasi-polynomial sample and computational complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds.

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arXiv2024

Online Learning of Halfspaces with Massart Noise

We study the task of online learning in the presence of Massart noise. Instead of assuming that the online adversary chooses an arbitrary sequence of labels, we assume that the context $\mathbf{x}$ is selected adversarially but the label $y$ presented to the learner disagrees with the ground-truth label of $\mathbf{x}$ with unknown probability at most $η$. We study the fundamental class of $γ$-margin linear classifiers and present a computationally efficient algorithm that achieves mistake bound $ηT + o(T)$. Our mistake bound is qualitatively tight for efficient algorithms: it is known that even in the offline setting achieving classification error better than $η$ requires super-polynomial time in the SQ model. We extend our online learning model to a $k$-arm contextual bandit setting where the rewards -- instead of satisfying commonly used realizability assumptions -- are consistent (in expectation) with some linear ranking function with weight vector $\mathbf{w}^\ast$. Given a list of contexts $\mathbf{x}_1,\ldots \mathbf{x}_k$, if $\mathbf{w}^*\cdot \mathbf{x}_i > \mathbf{w}^* \cdot \mathbf{x}_j$, the expected reward of action $i$ must be larger than that of $j$ by at least $Δ$. We use our Massart online learner to design an efficient bandit algorithm that obtains expected reward at least $(1-1/k)~ ΔT - o(T)$ bigger than choosing a random action at every round.

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arXiv2024

Super Non-singular Decompositions of Polynomials and their Application to Robustly Learning Low-degree PTFs

We study the efficient learnability of low-degree polynomial threshold functions (PTFs) in the presence of a constant fraction of adversarial corruptions. Our main algorithmic result is a polynomial-time PAC learning algorithm for this concept class in the strong contamination model under the Gaussian distribution with error guarantee $O_{d, c}(\text{opt}^{1-c})$, for any desired constant $c>0$, where $\text{opt}$ is the fraction of corruptions. In the strong contamination model, an omniscient adversary can arbitrarily corrupt an $\text{opt}$-fraction of the data points and their labels. This model generalizes the malicious noise model and the adversarial label noise model. Prior to our work, known polynomial-time algorithms in this corruption model (or even in the weaker adversarial label noise model) achieved error $\tilde{O}_d(\text{opt}^{1/(d+1)})$, which deteriorates significantly as a function of the degree $d$. Our algorithm employs an iterative approach inspired by localization techniques previously used in the context of learning linear threshold functions. Specifically, we use a robust perceptron algorithm to compute a good partial classifier and then iterate on the unclassified points. In order to achieve this, we need to take a set defined by a number of polynomial inequalities and partition it into several well-behaved subsets. To this end, we develop new polynomial decomposition techniques that may be of independent interest.

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arXiv2024

Robust Sparse Estimation for Gaussians with Optimal Error under Huber Contamination

We study Gaussian sparse estimation tasks in Huber's contamination model with a focus on mean estimation, PCA, and linear regression. For each of these tasks, we give the first sample and computationally efficient robust estimators with optimal error guarantees, within constant factors. All prior efficient algorithms for these tasks incur quantitatively suboptimal error. Concretely, for Gaussian robust $k$-sparse mean estimation on $\mathbb{R}^d$ with corruption rate $ε>0$, our algorithm has sample complexity $(k^2/ε^2)\mathrm{polylog}(d/ε)$, runs in sample polynomial time, and approximates the target mean within $\ell_2$-error $O(ε)$. Previous efficient algorithms inherently incur error $Ω(ε\sqrt{\log(1/ε)})$. At the technical level, we develop a novel multidimensional filtering method in the sparse regime that may find other applications.

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arXiv2024

Robust Second-Order Nonconvex Optimization and Its Application to Low Rank Matrix Sensing

Finding an approximate second-order stationary point (SOSP) is a well-studied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning. However, this problem is poorly understood in the presence of outliers, limiting the use of existing nonconvex algorithms in adversarial settings. In this paper, we study the problem of finding SOSPs in the strong contamination model, where a constant fraction of datapoints are arbitrarily corrupted. We introduce a general framework for efficiently finding an approximate SOSP with \emph{dimension-independent} accuracy guarantees, using $\widetilde{O}({D^2}/ε)$ samples where $D$ is the ambient dimension and $ε$ is the fraction of corrupted datapoints. As a concrete application of our framework, we apply it to the problem of low rank matrix sensing, developing efficient and provably robust algorithms that can tolerate corruptions in both the sensing matrices and the measurements. In addition, we establish a Statistical Query lower bound providing evidence that the quadratic dependence on $D$ in the sample complexity is necessary for computationally efficient algorithms.

math.OC↗

arXiv2024

SQ Lower Bounds for Non-Gaussian Component Analysis with Weaker Assumptions

We study the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model. Prior work developed a general methodology to prove SQ lower bounds for this task that have been applicable to a wide range of contexts. In particular, it was known that for any univariate distribution $A$ satisfying certain conditions, distinguishing between a standard multivariate Gaussian and a distribution that behaves like $A$ in a random hidden direction and like a standard Gaussian in the orthogonal complement, is SQ-hard. The required conditions were that (1) $A$ matches many low-order moments with the standard univariate Gaussian, and (2) the chi-squared norm of $A$ with respect to the standard Gaussian is finite. While the moment-matching condition is necessary for hardness, the chi-squared condition was only required for technical reasons. In this work, we establish that the latter condition is indeed not necessary. In particular, we prove near-optimal SQ lower bounds for NGCA under the moment-matching condition only. Our result naturally generalizes to the setting of a hidden subspace. Leveraging our general SQ lower bound, we obtain near-optimal SQ lower bounds for a range of concrete estimation tasks where existing techniques provide sub-optimal or even vacuous guarantees.

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arXiv2024

Statistical Query Lower Bounds for Learning Truncated Gaussians

We study the problem of estimating the mean of an identity covariance Gaussian in the truncated setting, in the regime when the truncation set comes from a low-complexity family $\mathcal{C}$ of sets. Specifically, for a fixed but unknown truncation set $S \subseteq \mathbb{R}^d$, we are given access to samples from the distribution $\mathcal{N}(\boldsymbol{ μ}, \mathbf{ I})$ truncated to the set $S$. The goal is to estimate $\boldsymbolμ$ within accuracy $ε>0$ in $\ell_2$-norm. Our main result is a Statistical Query (SQ) lower bound suggesting a super-polynomial information-computation gap for this task. In more detail, we show that the complexity of any SQ algorithm for this problem is $d^{\mathrm{poly}(1/ε)}$, even when the class $\mathcal{C}$ is simple so that $\mathrm{poly}(d/ε)$ samples information-theoretically suffice. Concretely, our SQ lower bound applies when $\mathcal{C}$ is a union of a bounded number of rectangles whose VC dimension and Gaussian surface are small. As a corollary of our construction, it also follows that the complexity of the previously known algorithm for this task is qualitatively best possible.

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arXiv2024

Robustly Learning Single-Index Models via Alignment Sharpness

We study the problem of learning Single-Index Models under the $L_2^2$ loss in the agnostic model. We give an efficient learning algorithm, achieving a constant factor approximation to the optimal loss, that succeeds under a range of distributions (including log-concave distributions) and a broad class of monotone and Lipschitz link functions. This is the first efficient constant factor approximate agnostic learner, even for Gaussian data and for any nontrivial class of link functions. Prior work for the case of unknown link function either works in the realizable setting or does not attain constant factor approximation. The main technical ingredient enabling our algorithm and analysis is a novel notion of a local error bound in optimization that we term alignment sharpness and that may be of broader interest.

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arXiv2024

How Does Unlabeled Data Provably Help Out-of-Distribution Detection?

Using unlabeled data to regularize the machine learning models has demonstrated promise for improving safety and reliability in detecting out-of-distribution (OOD) data. Harnessing the power of unlabeled in-the-wild data is non-trivial due to the heterogeneity of both in-distribution (ID) and OOD data. This lack of a clean set of OOD samples poses significant challenges in learning an optimal OOD classifier. Currently, there is a lack of research on formally understanding how unlabeled data helps OOD detection. This paper bridges the gap by introducing a new learning framework SAL (Separate And Learn) that offers both strong theoretical guarantees and empirical effectiveness. The framework separates candidate outliers from the unlabeled data and then trains an OOD classifier using the candidate outliers and the labeled ID data. Theoretically, we provide rigorous error bounds from the lens of separability and learnability, formally justifying the two components in our algorithm. Our theory shows that SAL can separate the candidate outliers with small error rates, which leads to a generalization guarantee for the learned OOD classifier. Empirically, SAL achieves state-of-the-art performance on common benchmarks, reinforcing our theoretical insights. Code is publicly available at https://github.com/deeplearning-wisc/sal.

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arXiv2023

Agnostically Learning Multi-index Models with Queries

We study the power of query access for the task of agnostic learning under the Gaussian distribution. In the agnostic model, no assumptions are made on the labels and the goal is to compute a hypothesis that is competitive with the {\em best-fit} function in a known class, i.e., it achieves error $\mathrm{opt}+ε$, where $\mathrm{opt}$ is the error of the best function in the class. We focus on a general family of Multi-Index Models (MIMs), which are $d$-variate functions that depend only on few relevant directions, i.e., have the form $g(\mathbf{W} \mathbf{x})$ for an unknown link function $g$ and a $k \times d$ matrix $\mathbf{W}$. Multi-index models cover a wide range of commonly studied function classes, including constant-depth neural networks with ReLU activations, and intersections of halfspaces. Our main result shows that query access gives significant runtime improvements over random examples for agnostically learning MIMs. Under standard regularity assumptions for the link function (namely, bounded variation or surface area), we give an agnostic query learner for MIMs with complexity $O(k)^{\mathrm{poly}(1/ε)} \; \mathrm{poly}(d) $. In contrast, algorithms that rely only on random examples inherently require $d^{\mathrm{poly}(1/ε)}$ samples and runtime, even for the basic problem of agnostically learning a single ReLU or a halfspace. Our algorithmic result establishes a strong computational separation between the agnostic PAC and the agnostic PAC+Query models under the Gaussian distribution. Prior to our work, no such separation was known -- even for the special case of agnostically learning a single halfspace, for which it was an open problem first posed by Feldman. Our results are enabled by a general dimension-reduction technique that leverages query access to estimate gradients of (a smoothed version of) the underlying label function.

cs.LG↗

arXiv2023

Clustering Mixtures of Bounded Covariance Distributions Under Optimal Separation

We study the clustering problem for mixtures of bounded covariance distributions, under a fine-grained separation assumption. Specifically, given samples from a $k$-component mixture distribution $D = \sum_{i =1}^k w_i P_i$, where each $w_i \ge α$ for some known parameter $α$, and each $P_i$ has unknown covariance $Σ_i \preceq σ^2_i \cdot I_d$ for some unknown $σ_i$, the goal is to cluster the samples assuming a pairwise mean separation in the order of $(σ_i+σ_j)/\sqrtα$ between every pair of components $P_i$ and $P_j$. Our contributions are as follows: For the special case of nearly uniform mixtures, we give the first poly-time algorithm for this clustering task. Prior work either required separation scaling with the maximum cluster standard deviation (i.e. $\max_i σ_i$) [DKK+22b] or required both additional structural assumptions and mean separation scaling as a large degree polynomial in $1/α$ [BKK22]. For general-weight mixtures, we point out that accurate clustering is information-theoretically impossible under our fine-grained mean separation assumptions. We introduce the notion of a clustering refinement -- a list of not-too-small subsets satisfying a similar separation, and which can be merged into a clustering approximating the ground truth -- and show that it is possible to efficiently compute an accurate clustering refinement of the samples. Furthermore, under a variant of the "no large sub-cluster'' condition from in prior work [BKK22], we show that our algorithm outputs an accurate clustering, not just a refinement, even for general-weight mixtures. As a corollary, we obtain efficient clustering algorithms for mixtures of well-conditioned high-dimensional log-concave distributions. Moreover, our algorithm is robust to $Ω(α)$-fraction of adversarial outliers.

cs.LG↗

arXiv2023

Near-Optimal Algorithms for Gaussians with Huber Contamination: Mean Estimation and Linear Regression

We study the fundamental problems of Gaussian mean estimation and linear regression with Gaussian covariates in the presence of Huber contamination. Our main contribution is the design of the first sample near-optimal and almost linear-time algorithms with optimal error guarantees for both of these problems. Specifically, for Gaussian robust mean estimation on $\mathbb{R}^d$ with contamination parameter $ε\in (0, ε_0)$ for a small absolute constant $ε_0$, we give an algorithm with sample complexity $n = \tilde{O}(d/ε^2)$ and almost linear runtime that approximates the target mean within $\ell_2$-error $O(ε)$. This improves on prior work that achieved this error guarantee with polynomially suboptimal sample and time complexity. For robust linear regression, we give the first algorithm with sample complexity $n = \tilde{O}(d/ε^2)$ and almost linear runtime that approximates the target regressor within $\ell_2$-error $O(ε)$. This is the first polynomial sample and time algorithm achieving the optimal error guarantee, answering an open question in the literature. At the technical level, we develop a methodology that yields almost-linear time algorithms for multi-directional filtering that may be of broader interest.

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arXiv2023

Testing Closeness of Multivariate Distributions via Ramsey Theory

We investigate the statistical task of closeness (or equivalence) testing for multidimensional distributions. Specifically, given sample access to two unknown distributions $\mathbf p, \mathbf q$ on $\mathbb R^d$, we want to distinguish between the case that $\mathbf p=\mathbf q$ versus $\|\mathbf p-\mathbf q\|_{A_k} > ε$, where $\|\mathbf p-\mathbf q\|_{A_k}$ denotes the generalized ${A}_k$ distance between $\mathbf p$ and $\mathbf q$ -- measuring the maximum discrepancy between the distributions over any collection of $k$ disjoint, axis-aligned rectangles. Our main result is the first closeness 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, we provide a computationally efficient closeness tester with sample complexity $O\left((k^{6/7}/ \mathrm{poly}_d(ε)) \log^d(k)\right)$. On the lower bound side, we establish a qualitatively matching sample complexity lower bound of $Ω(k^{6/7}/\mathrm{poly}(ε))$, even for $d=2$. These sample complexity bounds are surprising because the sample complexity of the problem in the univariate setting is $Θ(k^{4/5}/\mathrm{poly}(ε))$. This has the interesting consequence that the jump from one to two dimensions leads to a substantial increase in sample complexity, while increases beyond that do not. As a corollary of our general $A_k$ tester, we obtain $d_{\mathrm TV}$-closeness testers for pairs of $k$-histograms on $\mathbb R^d$ over a common unknown partition, and pairs of uniform distributions supported on the union of $k$ unknown disjoint axis-aligned rectangles. Both our algorithm and our lower bound make essential use of tools from Ramsey theory.

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