Abstract
To solve non-smooth convex optimization problems with a noisy gradient input, we analyze the global behavior ofsubgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of twoconvex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differentialinclusions 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 differentiableterm and a growth condition of the non-smooth term, our first main result shows almost sure weak convergenceof the trajectory process towards a minimizer of the objective function. Then, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergenceof the trajectory towards the minimum norm solution. We find an explicit tuning of this parameter when our objectivefunction satisfies a local error-bound inequality. We also provide a comprehensive complexity analysis by establishingseveral new pointwise and ergodic convergence rates in expectation for the convex and strongly convex case.