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In personalized Federated Learning, each member of a potentially large set ofagents aims to train a model minimizing its loss function averaged over itslocal data distribution. We study this problem under the lens of stochasticoptimization. Specifically, we introduce information-theoretic lower bounds onthe number of samples required from all agents to approximately minimize thegeneralization error of a fixed agent. We then provide strategies matchingthese lower bounds, in the all-for-one and all-for-all settings whererespectively one or all agents desire to minimize their own local function. Ourstrategies are based on a gradient filtering approach: provided prior knowledgeon some notions of distances or discrepancies between local data distributionsor functions, a given agent filters and aggregates stochastic gradientsreceived from other agents, in order to achieve an optimal bias-variancetrade-off.

Sample Optimality and All-for-all Strategies in Personalized Federated and Collaborative Learning