Fast Stochastic Composite Minimization and an Accelerated Frank-Wolfe Algorithm under Parallelization

TL;DR

This paper introduces a new accelerated stochastic optimization algorithm for convex functions, combining Bregman methods and Frank-Wolfe, achieving faster convergence especially with parallel computing.

Contribution

It proposes a novel Bregman-type accelerated algorithm for stochastic composite minimization and extends it to a parallelized Frank-Wolfe variant with improved convergence rates.

Findings

Achieves accelerated convergence in function values to a bounded region.

Demonstrates parallel Frank-Wolfe method with $ ilde{O}(1/ \sqrt{ ext{epsilon}})$ iteration complexity.

Validates the approach with synthetic numerical experiments.

Abstract

We consider the problem of minimizing the sum of two convex functions. One of those functions has Lipschitz-continuous gradients, and can be accessed via stochastic oracles, whereas the other is "simple". We provide a Bregman-type algorithm with accelerated convergence in function values to a ball containing the minimum. The radius of this ball depends on problem-dependent constants, including the variance of the stochastic oracle. We further show that this algorithmic setup naturally leads to a variant of Frank-Wolfe achieving acceleration under parallelization. More precisely, when minimizing a smooth convex function on a bounded domain, we show that one can achieve an $\epsilon$ primal-dual gap (in expectation) in $\tilde{O}(1/ \sqrt{\epsilon})$ iterations, by only accessing gradients of the original function and a linear maximization oracle with $O(1/\sqrt{\epsilon})$ computing units in parallel. We illustrate this fast convergence on synthetic numerical experiments.