Session II.2 - Continuous Optimization
Poster
An SDE perspective on stochastic convex optimization
Rodrigo Maulen
ENSICAEN, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this paper, we analyze the global and local behavior of gradient-like flows under stochastic errors towards the aim of solving convex optimization problems with noisy gradient input. We first study the unconstrained differentiable convex case, using a stochastic differential equation where the drift term is minus the gradient of the objective function and the diffusion term is either bounded or square-integrable. In this context, under Lipschitz continuity of the gradient, our first main result shows almost sure convergence of the objective and the trajectory process towards a minimizer of the objective function. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and (local) Polyak-Lojasiewicz case. The latter, which involves local analysis, is challenging and requires non-trivial arguments from measure theory.
Joint work with Jalal Fadili (ENSICAEN, France) and Hedy Attouch (Université de Montpellier, France).