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

Published: 31 Oct 2022, 18:00, Last Modified: 11 Jan 2023, 10:54NeurIPS 2022 AcceptReaders: Everyone
Keywords: Frank-Wolfe, Conditional Gradient, Acceleration, Randomized Smoothing, Convex Optimization
TL;DR: Fast Stochastic Composite Minimization and an Accelerated Frank-Wolfe Algorithm under Parallelization
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.
Supplementary Material: zip
12 Replies