The Sample Complexity of Simple Binary Hypothesis Testing

The sample complexity of simple binary hypothesis testing is the smallest number of i.i.d. samples required to distinguish between two distributions $p$ and $q$ in either: (i) the prior-free setting, with type-I error at most $\alpha$ and type-II error at most $\beta$; or (ii) the Bayesian setting, with Bayes error at most $\delta$ and prior distribution $(\alpha, 1-\alpha)$. This problem has only been studied when $\alpha = \beta$ (prior-free) or $\alpha = 1/2$ (Bayesian), and the sample complexity is known to be characterized by the Hellinger divergence between $p$ and $q$, up to multiplicative constants. In this paper, we derive a formula that characterizes the sample complexity (up to multiplicative constants that are independent of $p$, $q$, and all error parameters) for: (i) all $0 \le \alpha, \beta \le 1/8$ in the prior-free setting; and (ii) all $\delta \le \alpha/4$ in the Bayesian setting. In particular, the formula admits equivalent expressions in terms of certain divergences from the Jensen--Shannon and Hellinger families. The main technical result concerns an $f$-divergence inequality between members of the Jensen--Shannon and Hellinger families, which is proved by a combination of information-theoretic tools and case-by-case analyses. We explore applications of our results to robust and distributed (locally-private and communication-constrained) hypothesis testing.

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 $\epsilon>0$, our algorithm has sample complexity $(k^2/\epsilon^2)\mathrm{polylog}(d/\epsilon)$, runs in sample polynomial time, and approximates the target mean within $\ell_2$-error $O(\epsilon)$. Previous efficient algorithms inherently incur error $\Omega(\epsilon \sqrt{\log(1/\epsilon)})$. At the technical level, we develop a novel multidimensional filtering method in the sparse regime that may find other applications.

A Sub-Quadratic Time Algorithm for Robust Sparse Mean Estimation

We study the algorithmic problem of sparse mean estimation in the presence of adversarial outliers. Specifically, the algorithm observes a \emph{corrupted} set of samples from $\mathcal{N}(\mu,\mathbf{I}_d)$, where the unknown mean $\mu \in \mathbb{R}^d$ is constrained to be $k$-sparse. A series of prior works has developed efficient algorithms for robust sparse mean estimation with sample complexity $\mathrm{poly}(k,\log d, 1/\epsilon)$ and runtime $d^2 \mathrm{poly}(k,\log d,1/\epsilon)$, where $\epsilon$ is the fraction of contamination. In particular, the fastest runtime of existing algorithms is quadratic ($\Omega(d^2)$), which can be prohibitive in high dimensions. This quadratic barrier in the runtime stems from the reliance of these algorithms on the sample covariance matrix, which is of size $d^2$. Our main contribution is an algorithm for robust sparse mean estimation which runs in \emph{subquadratic} time using $\mathrm{poly}(k,\log d,1/\epsilon)$ samples. We also provide analogous results for robust sparse PCA. Our results build on algorithmic advances in detecting weak correlations, a generalized version of the light-bulb problem by Valiant.

Black-Box $k$-to-$1$-PCA Reductions: Theory and Applications

The $k$-principal component analysis ($k$-PCA) problem is a fundamental algorithmic primitive that is widely-used in data analysis and dimensionality reduction applications. In statistical settings, the goal of $k$-PCA is to identify a top eigenspace of the covariance matrix of a distribution, which we only have implicit access to via samples. Motivated by these implicit settings, we analyze black-box deflation methods as a framework for designing $k$-PCA algorithms, where we model access to the unknown target matrix via a black-box $1$-PCA oracle which returns an approximate top eigenvector, under two popular notions of approximation. Despite being arguably the most natural reduction-based approach to $k$-PCA algorithm design, such black-box methods, which recursively call a $1$-PCA oracle $k$ times, were previously poorly-understood. Our main contribution is significantly sharper bounds on the approximation parameter degradation of deflation methods for $k$-PCA. For a quadratic form notion of approximation we term ePCA (energy PCA), we show deflation methods suffer no parameter loss. For an alternative well-studied approximation notion we term cPCA (correlation PCA), we tightly characterize the parameter regimes where deflation methods are feasible. Moreover, we show that in all feasible regimes, $k$-cPCA deflation algorithms suffer no asymptotic parameter loss for any constant $k$. We apply our framework to obtain state-of-the-art $k$-PCA algorithms robust to dataset contamination, improving prior work both in sample complexity and approximation quality.

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 $\epsilon \in (0, \epsilon_0)$ for a small absolute constant $\epsilon_0$, we give an algorithm with sample complexity $n = \tilde{O}(d/\epsilon^2)$ and almost linear runtime that approximates the target mean within $\ell_2$-error $O(\epsilon)$. 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 $n = \tilde{O}(d/\epsilon^2)$ and almost linear runtime that approximates the target regressor within $\ell_2$-error $O(\epsilon)$. 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.

Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA

We study principal component analysis (PCA), where given a dataset in $\mathbb{R}^d$ from a distribution, the task is to find a unit vector $v$ that approximately maximizes the variance of the distribution after being projected along $v$. Despite being a classical task, standard estimators fail drastically if the data contains even a small fraction of outliers, motivating the problem of robust PCA. Recent work has developed computationally-efficient algorithms for robust PCA that either take super-linear time or have sub-optimal error guarantees. Our main contribution is to develop a nearly-linear time algorithm for robust PCA with near-optimal error guarantees. We also develop a single-pass streaming algorithm for robust PCA with memory usage nearly-linear in the dimension.

A Spectral Algorithm for List-Decodable Covariance Estimation in Relative Frobenius Norm

We study the problem of list-decodable Gaussian covariance estimation. Given a multiset $T$ of $n$ points in $\mathbb R^d$ such that an unknown $\alpha<1/2$ fraction of points in $T$ are i.i.d. samples from an unknown Gaussian $\mathcal{N}(\mu, \Sigma)$, the goal is to output a list of $O(1/\alpha)$ hypotheses at least one of which is close to $\Sigma$ in relative Frobenius norm. Our main result is a $\mathrm{poly}(d,1/\alpha)$ sample and time algorithm for this task that guarantees relative Frobenius norm error of $\mathrm{poly}(1/\alpha)$. Importantly, our algorithm relies purely on spectral techniques. As a corollary, we obtain an efficient spectral algorithm for robust partial clustering of Gaussian mixture models (GMMs) -- a key ingredient in the recent work of [BDJ+22] on robustly learning arbitrary GMMs. Combined with the other components of [BDJ+22], our new method yields the first Sum-of-Squares-free algorithm for robustly learning GMMs. At the technical level, we develop a novel multi-filtering method for list-decodable covariance estimation that may be useful in other settings.

Simple Binary Hypothesis Testing under Local Differential Privacy and Communication Constraints

We study simple binary hypothesis testing under both local differential privacy (LDP) and communication constraints. We qualify our results as either minimax optimal or instance optimal: the former hold for the set of distribution pairs with prescribed Hellinger divergence and total variation distance, whereas the latter hold for specific distribution pairs. For the sample complexity of simple hypothesis testing under pure LDP constraints, we establish instance-optimal bounds for distributions with binary support; minimax-optimal bounds for general distributions; and (approximately) instance-optimal, computationally efficient algorithms for general distributions. When both privacy and communication constraints are present, we develop instance-optimal, computationally efficient algorithms that achieve the minimum possible sample complexity (up to universal constants). Our results on instance-optimal algorithms hinge on identifying the extreme points of the joint range set $\mathcal A$ of two distributions $p$ and $q$, defined as $\mathcal A := \{(\mathbf T p, \mathbf T q) | \mathbf T \in \mathcal C\}$, where $\mathcal C$ is the set of channels characterizing the constraints.

Outlier-Robust Sparse Mean Estimation for Heavy-Tailed Distributions

We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity. Specifically, given a small number of corrupted samples from a high-dimensional heavy-tailed distribution whose mean $\mu$ is guaranteed to be sparse, the goal is to efficiently compute a hypothesis that accurately approximates $\mu$ with high probability. Prior work had obtained efficient algorithms for robust sparse mean estimation of light-tailed distributions. In this work, we give the first sample-efficient and polynomial-time robust sparse mean estimator for heavy-tailed distributions under mild moment assumptions. Our algorithm achieves the optimal asymptotic error using a number of samples scaling logarithmically with the ambient dimension. Importantly, the sample complexity of our method is optimal as a function of the failure probability $\tau$, having an additive $\log(1/\tau)$ dependence. Our algorithm leverages the stability-based approach from the algorithmic robust statistics literature, with crucial (and necessary) adaptations required in our setting. Our analysis may be of independent interest, involving the delicate design of a (non-spectral) decomposition for positive semi-definite matrices satisfying certain sparsity properties.

Gaussian Mean Testing Made Simple

We study the following fundamental hypothesis testing problem, which we term Gaussian mean testing. Given i.i.d. samples from a distribution $p$ on $\mathbb{R}^d$, the task is to distinguish, with high probability, between the following cases: (i) $p$ is the standard Gaussian distribution, $\mathcal{N}(0,I_d)$, and (ii) $p$ is a Gaussian $\mathcal{N}(\mu,\Sigma)$ for some unknown covariance $\Sigma$ and mean $\mu \in \mathbb{R}^d$ satisfying $\|\mu\|_2 \geq \epsilon$. Recent work gave an algorithm for this testing problem with the optimal sample complexity of $\Theta(\sqrt{d}/\epsilon^2)$. Both the previous algorithm and its analysis are quite complicated. Here we give an extremely simple algorithm for Gaussian mean testing with a one-page analysis. Our algorithm is sample optimal and runs in sample linear time.