Analysis of stochastic gradient descent in continuous time

Abstract Stochastic gradient descent is an optimisation method that combines classical with random subsampling within the target functional. In this work, we introduce stochastic process as a continuous-time representation of descent. The dynamical system coupled Markov living on finite state space. system—a flow—represents part, space represents subsampling. Processes type are, for instance, used to model clonal populations in fluctuating environments. After introducing it, study theoretical properties process: We show it converges weakly flow respect full function, learning rate approaches zero. give conditions under which constant exponentially ergodic Wasserstein sense. Then case, where goes zero sufficiently slowly and single functions are strongly convex. point mass concentrated global minimum function; indicating consistency method. conclude after discussion discretisation strategies numerical experiments.