From error bounds to the complexity of first-order descent methods for convex functions

• 成果类型: Article
• 署名作者: Bolte, Jerome; Trong Phong Nguyen; Peypouquet, Juan; Suter, Bruce W.
• 署名单位: Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics; Universidad Tecnica Federico Santa Maria; Universidad Tecnica Federico Santa Maria; United States Department of Defense; United States Air Force; US Air Force Research Laboratory; Yamaguchi University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1091-6
• 发表日期: 2017
• 页码: 471-507
• 关键词: thresholding algorithm; Asymptotic convergence; projection algorithms; polynomial systems; optimization; inequalities; minimization; STABILITY

摘要

This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-Aojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with Holderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple method: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with regularization.