Daniel Lacker (Columbia): Delocalization of bias in high-dimensional Langevin Monte Carlo

The unadjusted Langevin algorithm is widely used for sampling from complex high-dimensional distributions. It is well known to be biased, with the bias typically scaling linearly with the dimension when measured in squared Wasserstein distance. However, a recent paper of Chen, Cheng, Niles-Weed, and Weare identified an intriguing new "delocalization" phenomenon: For certain classes of target distributions, they proved that the bias between low-dimensional marginals scales only with the lower dimension, not the full dimension. This talk will explain this new phenomenon and our recent results which strengthen those of Chen et al. in several directions. The proofs are based on a hierarchical analysis of the marginal relative entropies (a.k.a. KL-divergence), inspired by our recent work on propagation of chaos for mean field models. A byproduct of our analysis, of independent interest, is a new kind of convergence estimate for low-dimensional marginals of continuous-time Langevin diffusions. Based on joint work with Fuzhong Zhou.