2018/07/20 by Haihao Lu, Lu, Haihao, Robert M. Freund +1 · 1 citation
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.1807.07680
openalex publication_date 2018/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
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 generalized stochastic Frank-Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of ``linear prediction'' property (formally defined in the paper), which is typically present loss minimization problems in practice. Our method also introduces the notion of a ``substitute gradient'' that is a not-necessarily-unbiased sample of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank-Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank-Wolfe methods consistent with the theory developed herein.