No-regret dynamics in the Fenchel game: a unified framework for algorithmic convex optimization

• 成果类型: Article
• 署名作者: Wang, Jun-Kun; Abernethy, Jacob; Levy, Kfir Y.
• 署名单位: Yale University; University System of Georgia; Georgia Institute of Technology; Technion Israel Institute of Technology
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01976-y
• 发表日期: 2024
• 页码: 203-268
• 关键词: variational-inequalities; 1st-order methods; splitting method; point algorithm; online; CONVERGENCE; gradient; rates

摘要

We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min-max game in a sequential fashion, we can consider a range of strategies for each of the two-players who must select their actions one after the other. A common choice for these strategies are so-called no-regret learning algorithms, and we describe a number of such and prove bounds on their regret. We then show that many classical first-order methods for convex optimization-including average-iterate gradient descent, the Frank-Wolfe algorithm, Nesterov's acceleration methods, the accelerated proximal method-can be interpreted as special cases of our framework as long as each player makes the correct choice of no-regret strategy. Proving convergence rates in this framework becomes very straightforward, as they follow from plugging in the appropriate known regret bounds. Our framework also gives rise to a number of new first-order methods for special cases of convex optimization that were not previously known.