2025/11/06 by Jianfeng Lu, Lu, Jianfeng, Xuda Ye +3 · 1 citation
Computer Science · Decision Sciences · Mathematics · #60H35 #62F15 #65C30 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Numerical Analysis (math.NA) #Simulation Techniques and Applications #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2511.04413
openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28
This paper considers mean square error (MSE) analysis for stochastic gradient sampling algorithms applied to underdamped Langevin dynamics under a global convexity assumption. A novel discrete Poisson equation framework is developed to bound the time-averaged sampling error. For the Stochastic Gradient UBU (SG-UBU) sampler, we derive an explicit MSE bound and establish that the numerical bias exhibits first-order convergence with respect to the step size h, with the leading error coefficient proportional to the variance of the stochastic gradient. The analysis is further extended to variance-reduced algorithms for finite-sum potentials, specifically the SVRG-UBU and SAGA-UBU methods. For these algorithms, we identify a phase transition phenomenon whereby the convergence rate of the numerical bias shifts from first to second order as the step size decreases below a critical threshold. Theoretical findings are validated by numerical experiments. In addition, the analysis provides a practical empirical criterion for selecting between the mini-batch SG-UBU and SVRG-UBU samplers to achieve optimal computational efficiency.