Robustness of Accelerated First-Order Algorithms for Strongly Convex Optimization Problems

• 成果类型: Article
• 署名作者: Mohammadi, Hesameddin; Razaviyayn, Meisam; Jovanovic, Mihailo R.
• 署名单位: University of Southern California; University of Southern California
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.3008297
• 发表日期: 2021
• 页码: 2480-2495
• 关键词: acceleration; CONVERGENCE; optimization; Robustness; Noise measurement; Heuristic algorithms; Upper bound; Accelerated first-order algorithms; Consensus networks; control for optimization; Convex Optimization; integral quadratic constraints; Linear matrix inequalities (LMIs); noise amplification; second-order moments; Semidefinite programming

摘要

We study the robustness of accelerated first-order algorithms to stochastic uncertainties in gradient evaluation. Specifically, for unconstrained, smooth, strongly convex optimization problems, we examine the mean-squared error in the optimization variable when the iterates are perturbed by additive white noise. This type of uncertainty may arise in situations where an approximation of the gradient is sought through measurements of a real system or in a distributed computation over a network. Even though the underlying dynamics of first-order algorithms for this class of problems are nonlinear, we establish upper bounds on the mean-squared deviation from the optimal solution that are tight up to constant factors. Our analysis quantifies fundamental tradeoffs between noise amplification and convergence rates obtained via any acceleration scheme similar to Nesterov's or heavy-ball methods. To gain additional analytical insight, for strongly convex quadratic problems, we explicitly evaluate the steady-state variance of the optimization variable in terms of the eigenvalues of the Hessian of the objective function. We demonstrate that the entire spectrum of the Hessian, rather than just the extreme eigenvalues, influences robustness of noisy algorithms. We specialize this result to the problem of distributed averaging over undirected networks and examine the role of network size and topology on the robustness of noisy accelerated algorithms.