COLT 2023 论文列表

We consider the question of estimating multi-dimensional Gaussian mixtures (GM) with com- pactly supported or subgaussian mixing distributions. Minimax estimation rate for this class (under Hellinger, TV and KL divergences) is a long-standing open question, even for dimension one. In this paper we characterize this rate (in all dimensions) in terms of the metric entropy of the class. Such characterizations originate from seminal works of Le Cam (1973); Birge ́ (1983); Haussler and Opper (1997); Yang and Barron (1999). However, for GMs a key ingredient missing from earlier work (and widely sought-after) is a comparison result showing that the KL and the squared Hellinger distance are within a constant multiple of each other uniformly over the class. Our main technical contribution is in showing this fact, from which we derive entropy characterization for estimation rate under Hellinger and KL. Interestingly, the sequential (online learning) estimation rate is characterized by the global entropy, while the single-step (batch) rate corresponds to local entropy, paralleling a similar recent discovery for the case of Gaussian sequence model in a pair of works Neykov (2022); Mourtada (2023). Additionally, since Hellinger is a proper metric, our comparison shows that GMs under KL satisfy a version of triangle inequality (with a multiplicative constant), implying that proper and improper estimation rates coincide.

We consider the problem of empirical Bayes estimation for (multivariate) Poisson means. Existing solutions that have been shown theoretically optimal for minimizing the regret (excess risk over the Bayesian oracle that knows the prior) have several shortcomings. For example, the classical Robbins estimator does not retain the monotonicity property of the Bayes estimator and performs poorly under moderate sample size. Estimators based on the minimum distance and non-parametric maximum likelihood (NPMLE) methods correct these issues, but are computationally expensive with complexity growing exponentially with dimension. Extending the approach of Barbehenn andZhao (2022), in this work we construct monotone estimators based on empirical risk minimization (ERM) that retain similar theoretical guarantees and can be computed much more efficiently. Adapting the idea of offset Rademacher complexity Liang et al. (2015) to the non-standard loss and function class in empirical Bayes, we show that the shape-constrained ERM estimator attains the minimax regret within constant factors in one dimension and within logarithmic factors in multiple dimensions.

The goal of this note is to present a modification of the popular median of means estimator that achieves sub-Gaussian deviation bounds with nearly optimal constants under minimal assumptions on the underlying distribution. We build on a recent work on the topic and prove that desired guarantees can be attained under weaker requirements.

We consider the sparse moment problem of learning a $k$-spike mixture in high-dimensional space from its noisy moment information in any dimension. We measure the accuracy of the learned mixtures using transportation distance. Previous algorithms either assume certain separation assumptions, use more recovery moments, or run in (super) exponential time. Our algorithm for the 1-dimensional problem (also called the sparse Hausdorff moment problem) is a robust version of the classic Prony’s method, and our contribution mainly lies in the analysis. We adopt a global and much tighter analysis than previous work (which analyzes the perturbation of the intermediate results of Prony’s method). A useful technical ingredient is a connection between the linear system defined by the Vandermonde matrix and the Schur polynomial, which allows us to provide tight perturbation bound independent of the separation and may be useful in other contexts. To tackle the high-dimensional problem, we first solve the 2-dimensional problem by extending the 1-dimensional algorithm and analysis to complex numbers. Our algorithm for the high-dimensional case determines the coordinates of each spike by aligning a 1d projection of the mixture to a random vector and a set of 2d projections of the mixture. Our results have applications to learning topic models and Gaussian mixtures, implying improved sample complexity results or running time over prior work.

We demonstrate the first algorithms for the problem of regression for generalized linear models (GLMs) in the presence of additive oblivious noise. We assume we have sample access to examples $(x, y)$ where $y$ is a noisy measurement of $g(w^* \cdot x)$. In particular, $y = g(w^* \cdot x) + \xi + \eps$ where $\xi$ is the oblivious noise drawn independently of $x$, satisfying $\Pr[\xi = 0] \geq o(1)$, and $\eps \sim \cN(0, \sigma^2)$. Our goal is to accurately recover a function $g(w \cdot x)$ with arbitrarily small error when compared to the true values $g(w^* \cdot x)$, rather than the noisy measurements $y$. We present an algorithm that tackles the problem in its most general distribution-independent setting, where the solution may not be identifiable. The algorithm is designed to return the solution if it is identifiable, and otherwise return a small list of candidates, one of which is close to the true solution. Furthermore, we characterize a necessary and sufficient condition for identifiability, which holds in broad settings. The problem is identifiable when the quantile at which $\xi + \eps = 0$ is known, or when the family of hypotheses does not contain candidates that are nearly equal to a translated $g(w^* \cdot x) + A$ for some real number $A$, while also having large error when compared to $g(w^* \cdot x)$. This is the first result for GLM regression which can handle more than half the samples being arbitrarily corrupted. Prior work focused largely on the setting of linear regression with oblivious noise, and giving algorithms under more restrictive assumptions.

We study the task of differentially private online convex optimization (OCO). In the online setting, the release of each distinct decision or iterate carries with it the potential for privacy loss. To limit such privacy leakage, we design an optimization-based OCO algorithm that explicitly limits the number of switches via objective perturbation and rejection sampling. This improves over known results in multiple aspects: an optimal leading-order regret term, in being efficiently implementable without requiring log-concave sampling subroutines, and in matching the non-private regret bound for sub-constant regimes of privacy parameters. Leveraging the fact that the algorithm is designed to explicitly minimize the number of switches of decisions, we show that the algorithm also obtains optimal regret bounds in the lazy OCO setting, where the learner is constrained to perform a limited number of switches. In addition, for one- and two-dimensional decision sets, we present a novel approach for differentially private online Lipschitz learning, where the loss functions are Lipschitz but not necessarily convex, that achieves the optimal regret bound matching known lower bounds.

We show that convex-concave Lipschitz stochastic saddle point problems (also known as stochastic minimax optimization) can be solved under the constraint of $(\epsilon,\delta)$-differential privacy with \emph{strong (primal-dual) gap} rate of $\tilde O\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{n\epsilon}\big)$, where $n$ is the dataset size and $d$ is the dimension of the problem. This rate is nearly optimal, based on existing lower bounds in differentially private stochastic optimization. Specifically, we prove a tight upper bound on the strong gap via novel implementation and analysis of the recursive regularization technique repurposed for saddle point problems. We show that this rate can be attained with $O\big(\min\big\{\frac{n^2\epsilon^{1.5}}{\sqrt{d}}, n^{3/2}\big\}\big)$ gradient complexity, and $\tilde{O}(n)$ gradient complexity if the loss function is smooth. As a byproduct of our method, we develop a general algorithm that, given a black-box access to a subroutine satisfying a certain $\alpha$ primal-dual accuracy guarantee with respect to the empirical objective, gives a solution to the stochastic saddle point problem with a strong gap of $\tilde{O}(\alpha+\frac{1}{\sqrt{n}})$. We show that this $\alpha$-accuracy condition is satisfied by standard algorithms for the empirical saddle point problem such as the proximal point method and the stochastic gradient descent ascent algorithm. Finally, to emphasize the importance of the strong gap as a convergence criterion compared to the weaker notion of primal-dual gap, commonly known as the \emph{weak gap}, we show that even for simple problems it is possible for an algorithm to have zero weak gap and suffer from $\Omega(1)$ strong gap. We also show that there exists a fundamental tradeoff between stability and accuracy. Specifically, we show that any $\Delta$-stable algorithm has empirical gap $\Omega\big(\frac{1}{\Delta n}\big)$, and that this bound is tight. This result also holds also more specifically for empirical risk minimization problems and may be of independent interest.

We study the complexity of optimizing nonsmooth nonconvex Lipschitz functions by producing $(\delta,\epsilon)$-Goldstein stationary points. Several recent works have presented randomized algorithms that produce such points using $\widetilde{O}(\delta^{-1}\epsilon^{-3})$ first-order oracle calls, independent of the dimension $d$. It has been an open problem as to whether a similar result can be obtained via a deterministic algorithm. We resolve this open problem, showing that randomization is necessary to obtain a dimension-free rate. In particular, we prove a lower bound of $\Omega(d)$ for any deterministic algorithm. Moreover, we show that unlike smooth or convex optimization, access to function values is required for any deterministic algorithm to halt within any finite time horizon.On the other hand, we prove that if the function is even slightly smooth, then the dimension-free rate of $\widetilde{O}(\delta^{-1}\epsilon^{-3})$ can be obtained by a deterministic algorithm with merely a logarithmic dependence on the smoothness parameter. Motivated by these findings, we turn to study the complexity of deterministically smoothing Lipschitz functions. Though there are well-known efficient black-box randomized smoothings, we start by showing that no such deterministic procedure can smooth functions in a meaningful manner (suitably defined), resolving an open question in the literature. We then bypass this impossibility result for the structured case of ReLU neural networks. To that end, in a practical “white-box” setting in which the optimizer is granted access to the network’s architecture, we propose a simple, dimension-free, deterministic smoothing of ReLU networks that provably preserves $(\delta,\epsilon)$-Goldstein stationary points. Our method applies to a variety of architectures of arbitrary depth, including ResNets and ConvNets.Combined with our algorithm for slightly-smooth functions, this yields the first deterministic, dimension-free algorithm for optimizing ReLU networks, circumventing our lower bound.

Planted Dense Subgraph (PDS) problem is a prototypical problem with a computational-statistical gap. It also exhibits an intriguing additional phenomenon: different tasks, such as detection or recovery, appear to have different computational limits. A detection-recovery gap for PDS was substantiated in the form of a precise conjecture given by Chen and Xu (2014) (based on the parameter values for which a convexified MLE succeeds), and then shown to hold for low-degree polynomial algorithms by Schramm and Wein (2022) and for MCMC algorithms for Ben Arous et al. (2020). In this paper we demonstrate that a slight variation of the Planted Clique Hypothesis with secret leakage (introduced in Brennan and Bresler (2020)), implies a detection-recovery gap for PDS. In the same vein, we also obtain a sharp lower bound for refutation, yielding a detection-refutation gap. Our methods build on the framework of Brennan and Bresler (2020) to construct average-case reductions mapping secret leakage Planted Clique to appropriate target problems.

Planted dense cycles are a type of latent structure that appears in many applications, such as small-world networks in social sciences and sequence assembly in computational biology. We consider a model where a dense cycle with expected bandwidth $n \tau$ and edge density $p$ is planted in an Erdős–Rényi graph $G(n,q)$. We characterize the computational thresholds for the associated detection and recovery problems for the class of low-degree polynomial algorithms. In particular, a gap exists between the two thresholds in a certain regime of parameters. For example, if $n^{-3/4} \ll \tau \ll n^{-1/2}$ and $p = C q = \Theta(1)$ for a constant $C>1$, the detection problem is computationally easy while the recovery problem is hard for low-degree algorithms.

We study the query complexity of geodesically convex (g-convex) optimization on a manifold. To isolate the effect of that manifold’s curvature, we primarily focus on hyperbolic spaces. In a variety of settings (smooth or not; strongly g-convex or not; high- or low-dimensional), known upper bounds worsen with curvature. It is natural to ask whether this is warranted, or an artifact.For many such settings, we propose a first set of lower bounds which indeed confirm that (negative) curvature is detrimental to complexity. To do so, we build on recent lower bounds (Hamilton and Moitra, 2021; Criscitiello and Boumal, 2022) for the particular case of smooth, strongly g-convex optimization. Using a number of techniques, we also secure lower bounds which capture dependence on condition number and optimality gap, which was not previously the case.We suspect these bounds are not optimal. We conjecture optimal ones, and support them with a matching lower bound for a class of algorithms which includes subgradient descent, and a lower bound for a related game. Lastly, to pinpoint the difficulty of proving lower bounds, we study how negative curvature influences (and sometimes obstructs) interpolation with g-convex functions.

We provide a simple convergence proof for AdaGrad optimizing non-convex objectives under only affine noise variance and bounded smoothness assumptions. The proof is essentially based on a novel auxiliary function $\xi$ that helps eliminate the complexity of handling the correlation between the numerator and denominator of AdaGrad’s update. Leveraging simple proofs, we are able to obtain tighter results than existing results [Faw et al 2002] and extend the analysis to several new and important cases. Specifically, for the over-parameterized regime, we show that AdaGrad needs only $\mathcal{O}(\frac{1}{\varepsilon^2})$ iterations to ensure the gradient norm smaller than $\varepsilon$, which matches the rate of SGD and significantly tighter than existing rates $\mathcal{O}(\frac{1}{\varepsilon^4})$ for AdaGrad. We then discard the bounded smoothness assumption, and consider a realistic assumption on smoothness called $(L_0,L_1)$-smooth condition, which allows local smoothness to grow with the gradient norm. Again based on the auxiliary function $\xi$, we prove that AdaGrad succeeds in converging under $(L_0,L_1)$-smooth condition as long as the learning rate is lower than a threshold. Interestingly, we further show that the requirement on learning rate under the $(L_0,L_1)$-smooth condition is necessary via proof by contradiction, in contrast with the case of uniform smoothness conditions where convergence is guaranteed regardless of learning rate choices. Together, our analyses broaden the understanding of AdaGrad and demonstrate the power of the new auxiliary function in the investigations of AdaGrad.

We consider contextual bandits with linear constraints (CBwLC), a variant of contextual bandits in which the algorithm consumes multiple resources subject to linear constraints on total consumption. This problem generalizes contextual bandits with knapsacks (CBwK), allowing for packing and covering constraints, as well as positive and negative resource consumption. We provide the first algorithm for CBwLC (or CBwK) that is based on regression oracles. The algorithm is simple, computationally efficient, and admits vanishing regret. It is statistically optimal for the variant of CBwK in which the algorithm must stop once some constraint is violated. Further, we provide the first vanishing-regret guarantees for CBwLC (or CBwK) that extend beyond the stochastic environment. We side-step strong impossibility results from prior work by identifying a weaker (and, arguably, fairer) benchmark to compare against. Our algorithm builds on LagrangeBwK (Immorlica et al., FOCS 2019), a Lagrangian-based technique for CBwK, and SquareCB (Foster and Rakhlin, ICML 2020), a regression-based technique for contextual bandits. Our analysis leverages the inherent modularity of both techniques.

In this paper, we address the stochastic contextual linear bandit problem, where a decision maker is provided a context (a random set of actions drawn from a distribution). The expected reward of each action is specified by the inner product of the action and an unknown parameter. The goal is to design an algorithm that learns to play as close as possible to the unknown optimal policy after a number of action plays. This problem is considered more challenging than the linear bandit problem, which can be viewed as a contextual bandit problem with a \emph{fixed} context. Surprisingly, in this paper, we show that the stochastic contextual problem can be solved as if it is a linear bandit problem. In particular, we establish a novel reduction framework that converts every stochastic contextual linear bandit instance to a linear bandit instance, when the context distribution is known. When the context distribution is unknown, we establish an algorithm that reduces the stochastic contextual instance to a sequence of linear bandit instances with small misspecifications and achieves nearly the same worst-case regret bound as the algorithm that solves the misspecified linear bandit instances. As a consequence, our results imply a $O(d\sqrt{T\log T})$ high-probability regret bound for contextual linear bandits, making progress in resolving an open problem in Li et al., 2019b, 2021. Our reduction framework opens up a new way to approach stochastic contextual linear bandit problems, and enables improved regret bounds in a number of instances including the batch setting, contextual bandits with misspecifications, contextual bandits with sparse unknown parameters, and contextual bandits with adversarial corruption.

We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of $e^{-f(x)}$ on a convex body $\mathcal{M}\subset\R^{n}$. We show that for distributions in the form of $e^{-\alpha^{\top}x}$ on a polytope with $m$ constraints, the convergence rate of a family of commonly-used integrators is independent of $\left\Vert \alpha\right\Vert _{2}$ and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of $\widetilde{O}\left(mn^{3}\right)$ to achieve $\epsilon$ total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form $e^{-f(x)}$ in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of \cite{kook2022sampling}, which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice.

We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution $\mu$, an $\varepsilon>0$, and access to sampling oracle(s) for a hidden distribution $\pi$, the goal in identity testing is to distinguish whether the two distributions $\mu$ and $\pi$ are identical or are at least $\varepsilon$-far apart. When there is only access to full samples from the hidden distribution $\pi$, it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various “conditional” sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the Coordinate Oracle, and provide a computational and statistical characterization of the identity testing problem in this new model.We prove that if an analytic property known as approximate tensorization of entropy holds for an $n$-dimensional visible distribution $\mu$, then there is an efficient identity testing algorithm for any hidden distribution $\pi$ using $\widetilde{O}(n/\varepsilon)$ queries to the Coordinate Oracle. Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of $n$-dimensional distributions, specifically sparse antiferromagnetic Ising models over $\{+1,-1\}^n$, we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless RP=NP. We complement our results with a matching $\Omega(n/\varepsilon)$ statistical lower bound for the sample complexity of identity testing in the $\coorora$ model.

Community detection is a fundamental problem in network science. In this paper, we consider community detection in hypergraphs drawn from the \emph{hypergraph stochastic block model} (HSBM), with a focus on exact community recovery. We study the performance of polynomial-time algorithms which operate on the \emph{similarity matrix} $W$, where $W_{ij}$ reports the number of hyperedges containing both $i$ and $j$. Under this information model, while the precise information-theoretic limit is unknown, Kim, Bandeira, and Goemans derived a sharp threshold up to which the natural min-bisection estimator on $W$ succeeds. As min-bisection is NP-hard in the worst case, they additionally proposed a semidefinite programming (SDP) relaxation and conjectured that it achieves the same recovery threshold as the min-bisection algorithm. In this paper, we confirm this conjecture. We also design a simple and highly efficient spectral algorithm with nearly linear runtime and show that it achieves the min-bisection threshold. Moreover, the spectral algorithm also succeeds in denser regimes and is considerably more efficient than previous approaches, establishing it as the method of choice. Our analysis of the spectral algorithm crucially relies on strong \emph{entrywise} bounds on the eigenvectors of $W$. Our bounds are inspired by the work of Abbe, Fan, Wang, and Zhong, who developed entrywise bounds for eigenvectors of symmetric matrices with independent entries. Despite the complex dependency structure in similarity matrices, we prove similar entrywise guarantees.

Matrix completion is the study of recovering an underlying matrix from a sparse subset of noisy observations. Traditionally, it is assumed that the entries of the matrix are “missing completely at random” (MCAR), i.e., each entry is revealed at random, independent of everything else, with uniform probability. This is likely unrealistic due to the presence of “latent confounders”, i.e., unobserved factors that determine both the entries of the underlying matrix and the missingness pattern in the observed matrix. For example, in the context of movie recommender systems—a canonical application for matrix completion—a user who vehemently dislikes horror films is unlikely to ever watch horror films. In general, these confounders yield “missing not at random” (MNAR) data, which can severely impact any inference procedure that does not correct for this bias. We develop a formal causal model for matrix completion through the language of potential outcomes, and provide novel identification arguments for a variety of causal estimands of interest. We design a procedure, which we call “synthetic nearest neighbors” (SNN), to estimate these causal estimands. We prove finite-sample consistency and asymptotic normality of our estimator. Our analysis also leads to new theoretical results for the matrix completion literature. In particular, we establish entry-wise, i.e., max-norm, finite-sample consistency and asymptotic normality results for matrix completion with MNAR data. As a special case, this also provides entry-wise bounds for matrix completion with MCAR data. Across simulated and real data, we demonstrate the efficacy of our proposed estimator.