Stochastic proximal subgradient descent oscillates in the vicinity of its accumulation set

We analyze the stochastic proximal subgradient descent in the case where the objective functions are path differentiable and verify a Sard-type condition. While the accumulation set may not be reduced to unique point, we show that the time spent by the iterates to move from one accumulation point to another goes to infinity. An oscillation-type behavior of the drift is established. These results show a strong stability property of the proximal subgradient descent. Using the theory of closed measures, Bolte et al. (2020) established this type of behavior for the deterministic subgradient descent. Our technique of proof relies on the classical works on stochastic approximation of differential inclusions, which allows us to extend results in the deterministic case to a stochastic and proximal setting, as well as to treat these different cases in a unified manner.