We consider a class of stochastic gradient optimization schemes. Assuming that the objective function is strongly convex, we prove weak error estimates which are uniform in time for the error between the solution of the numerical scheme, and the solutions of continuous-time modified (or high-resolution) differential equations at first and second orders, with respect to the time-step size. At first order, the modified equation is deterministic, whereas at second order the modified equation is stochastic and depends on a modified objective function. We go beyond existing results where the error estimates have been considered only on finite time intervals and were not uniform in time. This allows us to then provide a rigorous complexity analysis of the method in the large time and small time step size regimes.

翻译：我们考虑一类随机梯度优化方案。假设目标函数为强凸函数，我们证明了数值格式解与连续时间一阶和二阶修正（或高分辨率）微分方程解之间误差的弱误差估计，该估计关于时间步长一致且对时间一致。在一阶情况下，修正方程为确定性方程；而在二阶情况下，修正方程为随机方程且依赖于修正后的目标函数。我们突破了现有结果仅考虑有限时间区间且误差估计对时间不一致的限制。这使得我们能够在大时间与小时间步长机制下，对该方法进行严格的计算复杂度分析。