Interpolation Conditions for Linear Operators and Applications to Performance Estimation Problems
Year: 2024
Type: article
Abstract: . The performance estimation problem methodology makes it possible to determine the exact worst-case performance of an optimization method. In this work, we generalize this framework to first-order methods involving linear operators. This extension requires an explicit formulation of interpolation conditions for those linear operators. We consider the class of linear operators [Formula: see text], where matrix [Formula: see text] has bounded singular values, and the class of linear operators, where [Formula: see text] is symmetric and has bounded eigenvalues. We describe interpolation conditions for these classes, i.e., necessary and sufficient conditions that, given a list of pairs [Formula: see text], characterize the existence of a linear operator mapping [Formula: see text] to [Formula: see text] for all [Formula: see text]. Using these conditions, we first identify the exact worst-case behavior of the gradient method applied to the composed objective [Formula: see text], and observe that it always corresponds to [Formula: see text] being a scaling operator. We then investigate the Chambolle–Pock method applied to [Formula: see text], and improve the existing analysis to obtain a proof of the exact convergence rate of the primal-dual gap. In addition, we study how this method behaves on Lipschitz convex functions, and obtain a numerical convergence rate for the primal accuracy of the last iterate. We also show numerically that averaging iterates is beneficial in this setting.
Source: SIAM Journal on Optimization
DOI: 10.1137/23m1575391
Institution: UCLouvain
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4402279845