It can be shown that if a solution exists, it is unique.

In recent years, unbiased Markov Chain Monte Carlo via couplings (UMCMC) has emerged as a promising framework to remove bias from MCMC estimates, thus potentially allowing for early stopping, simplifying the convergence diagnostic process and facilitating parallelization (Glynn and Rhee, 2014; Jacob et al., 2020). In UMCMC, coupled chains are run for a random number of iterations (at least up to coalescence) and their values are combined to produce unbiased estimates. A natural question that arises is whether this class of estimates incurs a greater computational cost than conventional MCMC based on simple ergodic averages and to quantify this potential difference. Framing the question differently, one may ask whether it is possible to devise UMCMC methods with computational cost matching top performing MCMCs, while enjoying the above mentioned benefits. On a different line of research, various works showed how carefully designed blocked Gibbs Samplers (BGSs), i.e. Gibbs sampling schemes that update entire blocks of coordinates jointly, can achieve state-of-the-art performances for sampling from the posterior distributions of various challenging high-dimensional Bayesian models, such as non-nested models with crossed dependencies (Papaspiliopoulos et al., 2019, 2023). In particular, BGSs achieve linear computational costs in the number of parameters and observations in asymptotic regimes where both diverge to infinity.

The Stochastic Gradient Langevin Dynamics (SGLD), first introduced by Welling and Teh [25], has attracted a lot of attention in various areas [18, 26, 4]. The SGLD algorithm and its variants have fantastic performance when dealing with many practical sampling or optimization tasks. Recent decades have witnessed great development of theoretical research for SGLD, where most researchers focus on its discretization error, namely, the "distance" between the SGLD algorithm and the corresponding Langevin diffusion in terms of the time step (or learning rate) η [12, 18, 26, 16]. The SGLD itself can be regarded as a stochastic process and the ergodicity is also of great importance. So far, the justification of the geometric ergodicity of SGLD mostly relies on the strong convexity conditions, namely, the strong log-concaveness of the target distribution. In [4], under strong convexity settings, the authors considered the Synchronous coupling and established the geometric ergodicity of SGLD and some other numerical schemes in terms of Wasserstein-2 distance. However, the strong convexity assumption seems to limit the applicability of the result, and the ergodicity of the SGLD algorithm in a general setting and the existence of an invariant measure are still unclear. In our work, we aim to study the geometric ergodicity under locally nonconvex setting in this paper. The main technique we apply is reflection coupling [8], which was originally designed earlier to study the contraction property of many continuous SDEs.

Abstract: In this paper, we propose a weak approximation of the reflection coupling (RC) for stochastic differential equations (SDEs), and prove it converges weakly to the desired coupling. In contrast to the RC, the proposed approximate reflection coupling (ARC) need not take the hitting time of processes to the diagonal set into consideration and can be defined as the solution of some SDEs on the whole time interval. Therefore, ARC can work effectively against SDEs with different drift terms. As an application of ARC, an evaluation on the effectiveness of the stochastic gradient descent in a non-convex setting is also described. Abstract: The online optimization problem with non-convex loss functions over a closed convex set, coupled with a set of inequality (possibly non-convex) constraints is a challenging online learning problem.

In this paper, we propose a weak approximation of the reflection coupling (RC) for stochastic differential equations (SDEs), and prove it converges weakly to the desired coupling. In contrast to the RC, the proposed approximate reflection coupling (ARC) need not take the hitting time of processes to the diagonal set into consideration and can be defined as the solution of some SDEs on the whole time interval. Therefore, ARC can work effectively against SDEs with different drift terms. As an application of ARC, an evaluation on the effectiveness of the stochastic gradient descent in a non-convex setting is also described. For the sample size n, the step size η, and the batch size B, we derive uniform evaluations on the time with orders n -1, η 1/2, and ((n - B) / B (n - 1)), respectively.