Universal Generalization Guarantees for Wasserstein Distributionally Robust Models

Distributionally robust optimization has emerged as an attractive way to train robust machine learning models, capturing data uncertainty and distribution shifts. Recent statistical analyses have proved that robust models built from Wasserstein ambiguity sets have nice generalization guarantees, breaking the curse of dimensionality. However, these results are obtained in specific cases, at the cost of approximations, or under assumptions difficult to verify in practice. In contrast, we establish, in this article, exact generalization guarantees that cover all practical cases, including any transport cost function and any loss function, potentially non-convex and nonsmooth. For instance, our result applies to deep learning, without requiring restrictive assumptions. We achieve this result through a novel proof technique that combines nonsmooth analysis rationale with classical concentration results. Our approach is general enough to extend to the recent versions of Wasserstein/Sinkhorn distributionally robust problems that involve (double) regularizations.

Exact Generalization Guarantees for (Regularized) Wasserstein Distributionally Robust Models

Wasserstein distributionally robust estimators have emerged as powerful models for prediction and decision-making under uncertainty. These estimators provide attractive generalization guarantees: the robust objective obtained from the training distribution is an exact upper bound on the true risk with high probability. However, existing guarantees either suffer from the curse of dimensionality, are restricted to specific settings, or lead to spurious error terms. In this paper, we show that these generalization guarantees actually hold on general classes of models, do not suffer from the curse of dimensionality, and can even cover distribution shifts at testing. We also prove that these results carry over to the newly-introduced regularized versions of Wasserstein distributionally robust problems.

On the rate of convergence of Bregman proximal methods in constrained variational inequalities

We examine the last-iterate convergence rate of Bregman proximal methods - from mirror descent to mirror-prox - in constrained variational inequalities. Our analysis shows that the convergence speed of a given method depends sharply on the Legendre exponent of the underlying Bregman regularizer (Euclidean, entropic, or other), a notion that measures the growth rate of said regularizer near a solution. In particular, we show that boundary solutions exhibit a clear separation of regimes between methods with a zero and non-zero Legendre exponent respectively, with linear convergence for the former versus sublinear for the latter. This dichotomy becomes even more pronounced in linearly constrained problems where, specifically, Euclidean methods converge along sharp directions in a finite number of steps, compared to a linear rate for entropic methods.

Push--Pull with Device Sampling

We consider decentralized optimization problems in which a number of agents collaborate to minimize the average of their local functions by exchanging over an underlying communication graph. Specifically, we place ourselves in an asynchronous model where only a random portion of nodes perform computation at each iteration, while the information exchange can be conducted between all the nodes and in an asymmetric fashion. For this setting, we propose an algorithm that combines gradient tracking and variance reduction over the entire network. This enables each node to track the average of the gradients of the objective functions. Our theoretical analysis shows that the algorithm converges linearly, when the local objective functions are strongly convex, under mild connectivity conditions on the expected mixing matrices. In particular, our result does not require the mixing matrices to be doubly stochastic. In the experiments, we investigate a broadcast mechanism that transmits information from computing nodes to their neighbors, and confirm the linear convergence of our method on both synthetic and real-world datasets.

Federated Learning with Heterogeneous Data: A Superquantile Optimization Approach

We present a federated learning framework that is designed to robustly deliver good predictive performance across individual clients with heterogeneous data. The proposed approach hinges upon a superquantile-based learning objective that captures the tail statistics of the error distribution over heterogeneous clients. We present a stochastic training algorithm which interleaves differentially private client reweighting steps with federated averaging steps. The proposed algorithm is supported with finite time convergence guarantees that cover both convex and non-convex settings. Experimental results on benchmark datasets for federated learning demonstrate that our approach is competitive with classical ones in terms of average error and outperforms them in terms of tail statistics of the error.

The Last-Iterate Convergence Rate of Optimistic Mirror Descent in Stochastic Variational Inequalities

In this paper, we analyze the local convergence rate of optimistic mirror descent methods in stochastic variational inequalities, a class of optimization problems with important applications to learning theory and machine learning. Our analysis reveals an intricate relation between the algorithm's rate of convergence and the local geometry induced by the method's underlying Bregman function. We quantify this relation by means of the Legendre exponent, a notion that we introduce to measure the growth rate of the Bregman divergence relative to the ambient norm near a solution. We show that this exponent determines both the optimal step-size policy of the algorithm and the optimal rates attained, explaining in this way the differences observed for some popular Bregman functions (Euclidean projection, negative entropy, fractional power, etc.).

Optimization in Open Networks via Dual Averaging

In networks of autonomous agents (e.g., fleets of vehicles, scattered sensors), the problem of minimizing the sum of the agents' local functions has received a lot of interest. We tackle here this distributed optimization problem in the case of open networks when agents can join and leave the network at any time. Leveraging recent online optimization techniques, we propose and analyze the convergence of a decentralized asynchronous optimization method for open networks.

Multi-Agent Online Optimization with Delays: Asynchronicity, Adaptivity, and Optimism

Online learning has been successfully applied to many problems in which data are revealed over time. In this paper, we provide a general framework for studying multi-agent online learning problems in the presence of delays and asynchronicities. Specifically, we propose and analyze a class of adaptive dual averaging schemes in which agents only need to accumulate gradient feedback received from the whole system, without requiring any between-agent coordination. In the single-agent case, the adaptivity of the proposed method allows us to extend a range of existing results to problems with potentially unbounded delays between playing an action and receiving the corresponding feedback. In the multi-agent case, the situation is significantly more complicated because agents may not have access to a global clock to use as a reference point; to overcome this, we focus on the information that is available for producing each prediction rather than the actual delay associated with each feedback. This allows us to derive adaptive learning strategies with optimal regret bounds, at both the agent and network levels. Finally, we also analyze an "optimistic" variant of the proposed algorithm which is capable of exploiting the predictability of problems with a slower variation and leads to improved regret bounds.

Nonsmoothness in Machine Learning: specific structure, proximal identification, and applications

Nonsmoothness is often a curse for optimization; but it is sometimes a blessing, in particular for applications in machine learning. In this paper, we present the specific structure of nonsmooth optimization problems appearing in machine learning and illustrate how to leverage this structure in practice, for compression, acceleration, or dimension reduction. We pay a special attention to the presentation to make it concise and easily accessible, with both simple examples and general results.

First-order Optimization for Superquantile-based Supervised Learning

Classical supervised learning via empirical risk (or negative log-likelihood) minimization hinges upon the assumption that the testing distribution coincides with the training distribution. This assumption can be challenged in modern applications of machine learning in which learning machines may operate at prediction time with testing data whose distribution departs from the one of the training data. We revisit the superquantile regression method by proposing a first-order optimization algorithm to minimize a superquantile-based learning objective. The proposed algorithm is based on smoothing the superquantile function by infimal convolution. Promising numerical results illustrate the interest of the approach towards safer supervised learning.