IRON: Implicit Resolvent Optimization under Noise

Abstract

We study stochastic optimization from a joint continuous-discrete point of view. Starting from a second-order stochastic differential equation interpreted as a noisy accelerated gradient flow, we discretize the dynamics by a fully implicit Backward-Euler scheme. This leads to a resolvent, or proximal-type, update, computed in practice through Levenberg-Marquardt, Newton, or trust-region-type inner solves. The resulting method, denoted by $\text{IRON}_{\text{FI}}$, admits a Lyapunov mean-square recursion. The main conclusion is that increasing the implicit stepsize $\alpha$ improves the contraction factor and decreases the stationary mean-square error bound. Under sufficiently accurate inner solves, this bound scales as $O(1/\alpha)$; in particular, for large enough $\alpha$, the recursion is contractive and the stationary error bound vanishes as $\alpha\to\infty$. We establish the theory for smooth strongly convex objectives and provide a sharper quadratic analysis with an explicit stationary constant. The numerical experiments support the theory in the strongly convex case and illustrate the same qualitative behavior in nonconvex and learning settings. To the best of our knowledge, this fully implicit inertial-resolvent discretization, where the noise acts as an additive perturbation of the resolvent center and yields an $O(1/\alpha)$ stationary-MSE law, has not been isolated in this form before. The broader message is that discretization is not only an implementation choice: in stochastic optimization, the numerical integration rule can directly affect the long-time stability of the iterates under noise.