Continuous and Discrete-Time Analysis of Stochastic Gradient Descent for Convex and Non-Convex Functions

This paper proposes a thorough theoretical analysis of Stochastic Gradient Descent (SGD) with decreasing step sizes. First, we show that the recursion defining SGD can be provably approximated by solutions of a time inhomogeneous Stochastic Differential Equation (SDE) in a weak and strong sense. Then, motivated by recent analyses of deterministic and stochastic optimization methods by their continuous counterpart, we study the long-time convergence of the continuous processes at hand and establish non-asymptotic bounds. To that purpose, we develop new comparison techniques which we think are of independent interest. This continuous analysis allows us to develop an intuition on the convergence of SGD and, adapting the technique to the discrete setting, we show that the same results hold to the corresponding sequences. In our analysis, we notably obtain non-asymptotic bounds in the convex setting for SGD under weaker assumptions than the ones considered in previous works. Finally, we also establish finite time convergence results under various conditions, including relaxations of the famous {\L}ojasiewicz inequality, which can be applied to a class of non-convex functions.