In personalized federated learning, each member of a potentially large set of agents aims to train a model minimizing its loss function averaged over its local data distribution. We study this problem under the lens of stochastic optimization, focusing on a scenario with a large number of agents, that each possess very few data samples from their local data distribution. Specifically, we prove novel matching lower and upper bounds on the number of samples required from all agents to approximately minimize the generalization error of a fixed agent. We provide strategies matching these lower bounds, based on a gradient filtering approach: given prior knowledge on some notion of distance between local data distributions, agents filter and aggregate stochastic gradients received from other agents, in order to achieve an optimal bias-variance trade-off. Finally, we quantify the impact of using rough estimations of the distances between local distributions of agents, based on a very small number of local samples.

A major challenge in designing efficient statistical supervised learning algorithms is finding representations that perform well not only on available training samples but also on unseen data. While the study of representation learning has spurred much interest, most existing such approaches are heuristic; and very little is known about theoretical generalization guarantees. For example, the information bottleneck method seeks a good generalization by finding a minimal description of the input that is maximally informative about the label variable, where minimality and informativeness are both measured by Shannon's mutual information. In this paper, we establish a compressibility framework that allows us to derive upper bounds on the generalization error of a representation learning algorithm in terms of the "Minimum Description Length" (MDL) of the labels or the latent variables (representations). Rather than the mutual information between the encoder's input and the representation, which is often believed to reflect the algorithm's generalization capability in the related literature but in fact, falls short of doing so, our new bounds involve the "multi-letter" relative entropy between the distribution of the representations (or labels) of the training and test sets and a fixed prior.

Molecular Representation Learning (MRL) has emerged as a powerful tool for drug and materials discovery in a variety of tasks such as virtual screening and inverse design. While there has been a surge of interest in advancing modelcentric techniques, the influence of both data quantity and quality on molecular representations is not yet clearly understood within this field.