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

• URL: http://arxiv.org/abs/2205.12751v3
• Date: Mon, 25 Nov 2024 10:47:56 GMT
• Title: Fast Stochastic Composite Minimization and an Accelerated Frank-Wolfe Algorithm under Parallelization
• Authors: Benjamin Dubois-Taine, Francis Bach, Quentin Berthet, Adrien Taylor,
• Abstract summary: We consider the problem of minimizing the sum of two convex functions. One has Lipschitz-continuous gradients, and can be accessed via oracles, whereas the other is "simple" We show that one can achieve an $epsilon$ primaldual gap (in expectation) in $tildeO (1/ sqrtepsilon)$ iterations.
• Score: 7.197233473373693
• License:
• 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.

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