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Stochastic optimization methods have been hugely successful in makinglarge-scale optimization problems feasible when computing the full gradient iscomputationally prohibitive. Using the theory of modified equations fornumerical integrators, we propose a class of stochastic differential equationsthat approximate the dynamics of general stochastic optimization methods moreclosely than the original gradient flow. Analyzing a modified stochasticdifferential equation can reveal qualitative insights about the associatedoptimization method. Here, we study mean-square stability of the modifiedequation in the case of stochastic coordinate descent.

Backward error analysis and the qualitative behaviour of stochastic optimization algorithms: Application to stochastic coordinate descent