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Federated learning (FL) is a subfield of machine learning where multipleclients try to collaboratively learn a model over a network under communicationconstraints. We consider finite-sum federated optimization under a second-orderfunction similarity condition and strong convexity, and propose two newalgorithms: SVRP and Catalyzed SVRP. This second-order similarity condition hasgrown popular recently, and is satisfied in many applications includingdistributed statistical learning and differentially private empirical riskminimization. The first algorithm, SVRP, combines approximate stochasticproximal point evaluations, client sampling, and variance reduction. We showthat SVRP is communication efficient and achieves superior performance to manyexisting algorithms when function similarity is high enough. Our secondalgorithm, Catalyzed SVRP, is a Catalyst-accelerated variant of SVRP thatachieves even better performance and uniformly improves upon existingalgorithms for federated optimization under second-order similarity and strongconvexity. In the course of analyzing these algorithms, we provide a newanalysis of the Stochastic Proximal Point Method (SPPM) that might be ofindependent interest. Our analysis of SPPM is simple, allows for approximateproximal point evaluations, does not require any smoothness assumptions, andshows a clear benefit in communication complexity over ordinary distributedstochastic gradient descent.

Faster federated optimization under second-order similarity