Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces

Understand

To solve convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors.

• The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth.
• We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable.
• In this context, under Lipschitz's continuity of the differentiable term and a growth condition of the non-smooth term, our first main result shows almost sure weak convergence of the trajectory process towards a minimizer of the objective function.