Convergence of Langevin Monte Carlo in Chi-Square Divergence

We study sampling from a target distribution $\nu_* = e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm when the potential $f$ satisfies a strong dissipativity condition and it is first-order smooth with Lipschitz gradient. We prove that, initialized with a Gaussian that has sufficiently small variance, $\widetilde{\mathcal{O}}(\lambda d\epsilon^{-1})$ steps of the LMC algorithm are sufficient to reach $\epsilon$-neighborhood of the target in Chi-square divergence, where $\lambda$ is the log-Sobolev constant of $\nu_*$. Our results do not require warm-start to deal with exponential dimension dependency in Chi-square divergence at initialization. In particular, for strongly convex and first-order smooth potentials, we show that the LMC algorithm achieves the rate estimate $\widetilde{\mathcal{O}}(d\epsilon^{-1})$ which improves the previously known rates in this metric, under the same assumptions. Translating to other metrics, our result also recovers the best-known rate estimates in KL divergence, total variation and $2$-Wasserstein distance in the same setup. Finally, as we rely on the log-Sobolev inequality, our framework covers a wide range of non-convex potentials that are first-order smooth and that exhibit strong convexity outside of a compact region.