Sufficient convergence criterion for ergodic mean estimation in Langevin Monte Carlo

Develop a verifiable sufficient convergence criterion that guarantees the running mean of the loss trajectory (used to approximate ergodic expectations under the invariant distribution) has converged when applying the Unadjusted Langevin Algorithm or Stochastic Gradient Langevin Dynamics to approximate the Gibbs posterior. The criterion should ensure that the estimator based on the loss sequence (\hat{L}(h_{\beta_k t}, \mathbf{x}))_{t=0}^{T} reliably approximates \mathbb{E}_{h \sim G_{\beta_k}(\mathbf{x})}[\hat{L}(h, \mathbf{x})] without relying on ad hoc stopping rules.

Background

To approximate expectations under the Gibbs posterior in practice, the paper uses Langevin Monte Carlo methods (ULA and SGLD) and replaces posterior expectations with ergodic averages along the optimization trajectory. Specifically, the authors compute a running mean of the empirical loss along the sequence of iterates to estimate expectations in the invariant distribution.

However, in their experiments they do not have a principled, sufficient criterion to certify convergence of this ergodic mean estimator and instead employ a heuristic stopping rule when a slowly-varying running mean ceases to decrease. A rigorous, practical convergence criterion would strengthen the reliability of the bounds and the method’s applicability, particularly in the low-temperature interpolation regime where accurate posterior expectation estimates are critical.