Nonlocal Optimization: Imposing Structure on Optimization Problems by Relaxation
Abstract
In stochastic optimization, particularly in evolutionary computation and reinforcement learning, the optimization of a function $f: \Omega \to \mathbb{R}$ is often addressed through optimizing a socalled relaxation $\theta \in \Theta \mapsto \mathbb{E}_\theta(f)$ of $f$, where $\Theta$ resembles the parameters of a family of probability measures on $\Omega$. We investigate the structure of such relaxations by means of measure theory and Fourier analysis, enabling us to shed light on the success of many associated stochastic optimization methods. The main structural traits we derive and that allow fast and reliable optimization of relaxations are the consistency of optimal values of $f$, Lipschitzness of gradients, and convexity. We emphasize settings where $f$ itself is not differentiable or convex, e.g., in the presence of (stochastic) disturbance.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 arXiv:
 arXiv:2011.06064
 Bibcode:
 2020arXiv201106064M
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Neural and Evolutionary Computing;
 Statistics  Machine Learning;
 90C15 (Primary) 90C30;
 90C56 (Secondary)
 EPrint:
 19 pages, 1 figure, final version