Optimization on a Finer Scale: Bounded Local Subgradient Variation Perspective

We initiate the study of nonsmooth optimization problems under bounded local subgradient variation, which postulates bounded difference between (sub)gradients in small local regions around points, in either average or maximum sense. The resulting class of objective functions encapsulates the classes of objective functions traditionally studied in optimization, which are defined based on either Lipschitz continuity of the objective or H\"{o}lder/Lipschitz continuity of its gradient. Further, the defined class contains functions that are neither Lipschitz continuous nor have a H\"{o}lder continuous gradient. When restricted to the traditional classes of optimization problems, the parameters defining the studied classes lead to more fine-grained complexity bounds, recovering traditional oracle complexity bounds in the worst case but generally leading to lower oracle complexity for functions that are not ``worst case.'' Some highlights of our results are that: (i) it is possible to obtain complexity results for both convex and nonconvex problems with the (local or global) Lipschitz constant being replaced by a constant of local subgradient variation and (ii) mean width of the subdifferential set around the optima plays a role in the complexity of nonsmooth optimization, particularly in parallel settings. A consequence of (ii) is that for any error parameter $\epsilon > 0$, parallel oracle complexity of nonsmooth Lipschitz convex optimization is lower than its sequential oracle complexity by a factor $\tilde{\Omega}\big(\frac{1}{\epsilon}\big)$ whenever the objective function is piecewise linear with polynomially many pieces in the input size. This is particularly surprising as existing parallel complexity lower bounds are based on such classes of functions. The seeming contradiction is resolved by considering the region in which the algorithm is allowed to query the objective.

A Primal-Dual Algorithm for Faster Distributionally Robust Optimization

We consider the penalized distributionally robust optimization (DRO) problem with a closed, convex uncertainty set, a setting that encompasses the $f$-DRO, Wasserstein-DRO, and spectral/$L$-risk formulations used in practice. We present Drago, a stochastic primal-dual algorithm that achieves a state-of-the-art linear convergence rate on strongly convex-strongly concave DRO problems. The method combines both randomized and cyclic components with mini-batching, which effectively handles the unique asymmetric nature of the primal and dual problems in DRO. We support our theoretical results with numerical benchmarks in classification and regression.

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 $\widetilde{O}({D^2}/{\epsilon})$ samples where $D$ is the ambient dimension and $\epsilon$ 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.

Last Iterate Convergence of Incremental Methods and Applications in Continual Learning

Incremental gradient methods and incremental proximal methods are a fundamental class of optimization algorithms used for solving finite sum problems, broadly studied in the literature. Yet, when it comes to their convergence guarantees, nonasymptotic (first-order or proximal) oracle complexity bounds have been obtained fairly recently, almost exclusively applying to the average iterate. Motivated by applications in continual learning, we obtain the first convergence guarantees for the last iterate of both incremental gradient and incremental proximal methods, in general convex smooth (for both) and convex Lipschitz (for the proximal variants) settings. Our oracle complexity bounds for the last iterate nearly match (i.e., match up to a square-root-log or a log factor) the best known oracle complexity bounds for the average iterate, for both classes of methods. We further obtain generalizations of our results to weighted averaging of the iterates with increasing weights, which can be seen as interpolating between the last iterate and the average iterate guarantees. Additionally, we discuss how our results can be generalized to variants of studied incremental methods with permuted ordering of updates. Our results generalize last iterate guarantees for incremental methods compared to state of the art, as such results were previously known only for overparameterized linear models, which correspond to convex quadratic problems with infinitely many solutions.

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.

Variance Reduced Halpern Iteration for Finite-Sum Monotone Inclusions

Machine learning approaches relying on such criteria as adversarial robustness or multi-agent settings have raised the need for solving game-theoretic equilibrium problems. Of particular relevance to these applications are methods targeting finite-sum structure, which generically arises in empirical variants of learning problems in these contexts. Further, methods with computable approximation errors are highly desirable, as they provide verifiable exit criteria. Motivated by these applications, we study finite-sum monotone inclusion problems, which model broad classes of equilibrium problems. Our main contributions are variants of the classical Halpern iteration that employ variance reduction to obtain improved complexity guarantees in which $n$ component operators in the finite sum are ``on average'' either cocoercive or Lipschitz continuous and monotone, with parameter $L$. The resulting oracle complexity of our methods, which provide guarantees for the last iterate and for a (computable) operator norm residual, is $\widetilde{\mathcal{O}}( n + \sqrt{n}L\varepsilon^{-1})$, which improves upon existing methods by a factor up to $\sqrt{n}$. This constitutes the first variance reduction-type result for general finite-sum monotone inclusions and for more specific problems such as convex-concave optimization when operator norm residual is the optimality measure. We further argue that, up to poly-logarithmic factors, this complexity is unimprovable in the monotone Lipschitz setting; i.e., the provided result is near-optimal.

Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise

We study the problem of learning general (i.e., not necessarily homogeneous) halfspaces with Random Classification Noise under the Gaussian distribution. We establish nearly-matching algorithmic and Statistical Query (SQ) lower bound results revealing a surprising information-computation gap for this basic problem. Specifically, the sample complexity of this learning problem is $\widetilde{\Theta}(d/\epsilon)$, where $d$ is the dimension and $\epsilon$ is the excess error. Our positive result is a computationally efficient learning algorithm with sample complexity $\tilde{O}(d/\epsilon + d/(\max\{p, \epsilon\})^2)$, where $p$ quantifies the bias of the target halfspace. On the lower bound side, we show that any efficient SQ algorithm (or low-degree test) for the problem requires sample complexity at least $\Omega(d^{1/2}/(\max\{p, \epsilon\})^2)$. Our lower bound suggests that this quadratic dependence on $1/\epsilon$ is inherent for efficient algorithms.

Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise

We study the problem of PAC learning $\gamma$-margin halfspaces with Random Classification Noise. We establish an information-computation tradeoff suggesting an inherent gap between the sample complexity of the problem and the sample complexity of computationally efficient algorithms. Concretely, the sample complexity of the problem is $\widetilde{\Theta}(1/(\gamma^2 \epsilon))$. We start by giving a simple efficient algorithm with sample complexity $\widetilde{O}(1/(\gamma^2 \epsilon^2))$. Our main result is a lower bound for Statistical Query (SQ) algorithms and low-degree polynomial tests suggesting that the quadratic dependence on $1/\epsilon$ in the sample complexity is inherent for computationally efficient algorithms. Specifically, our results imply a lower bound of $\widetilde{\Omega}(1/(\gamma^{1/2} \epsilon^2))$ on the sample complexity of any efficient SQ learner or low-degree test.

Empirical Risk Minimization with Shuffled SGD: A Primal-Dual Perspective and Improved Bounds

Stochastic gradient descent (SGD) is perhaps the most prevalent optimization method in modern machine learning. Contrary to the empirical practice of sampling from the datasets without replacement and with (possible) reshuffling at each epoch, the theoretical counterpart of SGD usually relies on the assumption of sampling with replacement. It is only very recently that SGD with sampling without replacement -- shuffled SGD -- has been analyzed. For convex finite sum problems with $n$ components and under the $L$-smoothness assumption for each component function, there are matching upper and lower bounds, under sufficiently small -- $\mathcal{O}(\frac{1}{nL})$ -- step sizes. Yet those bounds appear too pessimistic -- in fact, the predicted performance is generally no better than for full gradient descent -- and do not agree with the empirical observations. In this work, to narrow the gap between the theory and practice of shuffled SGD, we sharpen the focus from general finite sum problems to empirical risk minimization with linear predictors. This allows us to take a primal-dual perspective and interpret shuffled SGD as a primal-dual method with cyclic coordinate updates on the dual side. Leveraging this perspective, we prove a fine-grained complexity bound that depends on the data matrix and is never worse than what is predicted by the existing bounds. Notably, our bound can predict much faster convergence than the existing analyses -- by a factor of the order of $\sqrt{n}$ in some cases. We empirically demonstrate that on common machine learning datasets our bound is indeed much tighter. We further show how to extend our analysis to convex nonsmooth problems, with similar improvements.

Robustly Learning a Single Neuron via Sharpness

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal $L_2^2$-error within a constant factor. Our algorithm applies under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.