On the cost of Bayesian posterior mean strategy for log-concave models
This work addresses the computational efficiency of Bayesian inference for statisticians and machine learning practitioners, but it is incremental as it builds on existing Langevin diffusion methods.
The paper tackles the problem of computing Bayesian estimators via Langevin Monte-Carlo approximations in log-concave models, establishing optimal numerical strategies and costs for approximating the true parameter with the Bayesian posterior mean, including quantitative bounds based on Poincaré constants and new results for weakly convex cases.
In this paper, we investigate the problem of computing Bayesian estimators using Langevin Monte-Carlo type approximation. The novelty of this paper is to consider together the statistical and numerical counterparts (in a general log-concave setting). More precisely, we address the following question: given $n$ observations in $\mathbb{R}^q$ distributed under an unknown probability $\mathbb{P}_{θ^\star}$ with $θ^\star \in \mathbb{R}^d$ , what is the optimal numerical strategy and its cost for the approximation of $θ^\star$ with the Bayesian posterior mean? To answer this question, we establish some quantitative statistical bounds related to the underlying Poincaré constant of the model and establish new results about the numerical approximation of Gibbs measures by Cesaro averages of Euler schemes of (over-damped) Langevin diffusions. These last results include in particular some quantitative controls in the weakly convex case based on new bounds on the solution of the related Poisson equation of the diffusion.