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

• URL: http://arxiv.org/abs/2309.02082v1
• Date: Tue, 5 Sep 2023 09:39:56 GMT
• Title: Backward error analysis and the qualitative behaviour of stochastic optimization algorithms: Application to stochastic coordinate descent
• Authors: Stefano Di Giovacchino, Desmond J. Higham, Konstantinos Zygalakis
• Abstract summary: We propose a class of differential equations that approximate the dynamics of general optimization methods more closely than the original gradient flow. We study the stability of the modified equation in the case of coordinate descent.
• Score: 1.534667887016089
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical integrators, we propose a class of stochastic differential equations that approximate the dynamics of general stochastic optimization methods more closely than the original gradient flow. Analyzing a modified stochastic differential equation can reveal qualitative insights about the associated optimization method. Here, we study mean-square stability of the modified equation in the case of stochastic coordinate descent.

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