Why random reshuffling beats stochastic gradient descent

• 成果类型: Article
• 署名作者: Gurbuzbalaban, M.; Ozdaglar, A.; Parrilo, P. A.
• 署名单位: Rutgers University System; Rutgers University New Brunswick; Massachusetts Institute of Technology (MIT)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-019-01440-w
• 发表日期: 2021
• 页码: 49-84
• 关键词: Approximation; algorithms; robust

摘要

We analyze the convergence rate of the random reshuffling (RR) method, which is a randomized first-order incremental algorithm for minimizing a finite sum of convex component functions. RR proceeds in cycles, picking a uniformly random order (permutation) and processing the component functions one at a time according to this order, i.e., at each cycle, each component function is sampled without replacement from the collection. Though RR has been numerically observed to outperform its with-replacement counterpart stochastic gradient descent (SGD), characterization of its convergence rate has been a long standing open question. In this paper, we answer this question by providing various convergence rate results for RR and variants when the sum function is strongly convex. We first focus on quadratic component functions and show that the expected distance of the iterates generated by RR with stepsize alpha k=Theta(1/ks) requiring adjusting the stepsize to the strong convexity constant). Our main result shows that when the component functions are quadratics or smooth (with a Lipschitz assumption on the Hessian matrices), RR with iterate averaging and a diminishing stepsize alpha k=Theta(1/ks) rate of SGD. Our analysis draws on the theory of Polyak-Ruppert averaging and relies on decoupling the dependent cycle gradient error into an independent term over cycles and another term dominated by alpha k2. This allows us to apply law of large numbers to an appropriately weighted version of the cycle gradient errors, where the weights depend on the stepsize. We also provide high probability convergence rate estimates that shows decay rate of different terms and allows us to propose a modification of RR with convergence rate O(1k2).