Stable Bounds on the Duality Gap of Separable Nonconvex Optimization Problems

• 成果类型: Article
• 署名作者: Kerdreux, Thomas; Colin, Igor; d'Aspremont, Alexandre
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - Institute for Information Sciences & Technologies (INS2I); Universite PSL; Ecole Normale Superieure (ENS); Huawei Technologies
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1291
• 发表日期: 2023
• 页码: 1044-1065
• 关键词: probability-inequalities; convex-programs

摘要

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland show that this produces an a priori bound on the duality gap of separable nonconvex optimization problems involving finite sums. This bound is highly conservative and depends on unstable quantities; we relax it in several directions to show that nonconvexity can have a much milder impact on finite sum minimization problems, such as empirical risk minimization and multitask classification. As a byproduct, we show a new version of Maurey's classical approximate Caratheodory lemma where we sample a significant fraction of the coefficients, without replacement, as well as a result on sampling constraints using an approximate Helly theorem, both of independent interest.