Double Randomized Underdamped Langevin with Dimension-Independent Convergence Guarantee

Part of Advances in Neural Information Processing Systems 36 (NeurIPS 2023) Main Conference Track

Authors

Yuanshi Liu, Cong Fang, Tong Zhang

Abstract

This paper focuses on the high-dimensional sampling of log-concave distributions with composite structures: $p^*(\mathrm{d}x)\propto \exp(-g(x)-f(x))\mathrm{d}x$. We develop a double randomization technique, which leads to a fast underdamped Langevin algorithm with a dimension-independent convergence guarantee. We prove that the algorithm enjoys an overall $\tilde{\mathcal{O}}\left(\frac{\left(\mathrm{tr}(H)\right)^{1/3}}{\epsilon^{2/3}}\right)$ iteration complexity to reach an $\epsilon$-tolerated sample whose distribution $p$ admits $W_2(p,p^*)\leq \epsilon$. Here, $H$ is an upper bound of the Hessian matrices for $f$ and does not explicitly depend on dimension $d$. For the posterior sampling over linear models with normalized data, we show a clear superiority of convergence rate which is dimension-free and outperforms the previous best-known results by a $d^{1/3}$ factor. The analysis to achieve a faster convergence rate brings new insights into high-dimensional sampling.