-
2025
En-Nebbazi Nassim
[abstract]
We consider a class of stochastic Heavy--Ball optimization schemes. Assuming that the objective function is strongly convex, we prove weak error estimate which is uniform in time for the error between the solution of the numerical scheme, and the solution of continuous-time modified (or high-resolution) differential equations at first order, with respect to the time-step size. At first order, the modified equation is deterministic. 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. We provide numerical experiments to illustrate the convergence result.
-
2025
Bréhier Charles-Édouard, Dambrine Marc et En-Nebbazi Nassim
[abstract]
We study a semi-implicit scheme which can be applied to minimize objective functions which are decomposed as the sum of a quadratic term and of a nonlinear function, perturbed by noise. We exhibit modified deterministic and stochastic modified equations which are employed to analyze the convergence of the algorithm in terms of strong and weak error estimates. We compare the results with those obtained for a standard explicit stochastic gradient optimization scheme.
-
2025
Bréhier Charles-Édouard, Dambrine Marc et En-Nebbazi Nassim
[abstract]
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. We provide numerical experiments to illustrate the convergence results.