2024/04/27 by Radu Ioan Boţ, Bot, Radu Ioan, Chiara Schindler +1 · 2 citations
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Nonlinear Differential Equations Analysis #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2404.17986
In our pursuit of finding a zero for a monotone and Lipschitz continuous operator M : \Rn → \Rn amidst noisy evaluations, we explore an associated differential equation within a stochastic framework, incorporating a correction term. We present a result establishing the existence and uniqueness of solutions for the stochastic differential equations under examination. Additionally, assuming that the diffusion term is square-integrable, we demonstrate the almost sure convergence of the trajectory process X(t) to a zero of M and of ‖M(X(t))‖ to 0 as t → +∞. Furthermore, we provide ergodic upper bounds and ergodic convergence rates in expectation for ‖M(X(t))‖2 and ⟨ M(X(t), X(t)-x^*⟩, where x^* is an arbitrary zero of the monotone operator. Subsequently, we apply these findings to a minimax problem. Finally, we analyze two temporal discretizations of the continuous-time models, resulting in stochastic variants of the Optimistic Gradient Descent Ascent and Extragradient methods, respectively, and assess their convergence properties.