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Generalized Stochastic Frank-Wolfe Algorithm with Stochastic "Substitute'' Gradient for Structured Convex Optimization

Haihao Lu (haihao***at***mit.edu)
Robert Freund (rfreund***at***mit.edu)

Abstract: The stochastic Frank-Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the guaranteed convergence rate for stochastic Frank-Wolfe compared to its deterministic counterpart. In this work, we present a new stochastic Frank-Wolfe method which closes this gap by introducing the notion of a "substitute'' gradient" that is a not-necessarily unbiased sample of the gradient. Moreover, we show that this new approach is equivalent to a randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of dual coordinate descent method in the primal space. When the regularizer is furthermore strongly convex, we show that the generalized stochastic Frank-Wolfe method as well as the randomized dual coordinate descent present linear convergence. These new results are benefited from the understanding that first-order methods can inherently minimize the primal-dual gap.

Keywords: Frank-Wolfe, conditional gradient, stochastic gradient, computational complexity

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Stochastic Programming

Category 3: Nonlinear Optimization (Other )

Citation: MIT Operations Research Center Working Paper, MIT, July 2018.

Download: [PDF]

Entry Submitted: 07/29/2018
Entry Accepted: 07/29/2018
Entry Last Modified: 09/04/2018

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