Lyapunov Analysis on Stochastic Primal-Dual Hybrid Gradient Method

In this paper, we study the convergence properties of Stochastic Primal-Dual Hybrid Gradient (SPDHG) methods for solving high-dimensional convex optimization problems under stochastic settings, which frequently arise in machine learning, statistics, and imaging science. Existing studies primarily focus on the deterministic PDHG, while the stochastic variant remains theoretically underexplored, especially in the presence of noise. To address this, we propose a novel analysis framework based on Lyapunov function tailored for the stochastic setting. Our approach enables the derivation of explicit global non-ergodic convergence rates and high probability performance guarantees without requiring second-order information. Empirically, we conduct extensive experiments on logistic regression task, demonstrating the robustness and efficiency of SPDHG.