Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence

• URL: http://arxiv.org/abs/2007.11612v4
• Date: Thu, 8 Jul 2021 06:45:09 GMT
• Title: Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence
• Authors: Murat A. Erdogdu, Rasa Hosseinzadeh, Matthew S. Zhang
• Abstract summary: We show that the rate estimate $widetildemathcalO(depsilon-1)$ improves the previously known rates in both of these metrics. In particular, for convex and firstorder smooth potentials, we show that LMC algorithm achieves the rate estimate $widetildemathcalO(depsilon-1)$ which improves the previously known rates in both of these metrics.
• Score: 8.873449722727026
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 a Lipschitz gradient. We prove that, initialized with a Gaussian random vector that has sufficiently small variance, iterating the LMC algorithm for $\widetilde{\mathcal{O}}(\lambda^2 d\epsilon^{-1})$ steps is sufficient to reach $\epsilon$-neighborhood of the target in both Chi-squared and Renyi divergence, where $\lambda$ is the logarithmic Sobolev constant of $\nu_*$. Our results do not require warm-start to deal with the exponential dimension dependency in Chi-squared 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 both of these metrics, under the same assumptions. Translating this rate to other metrics, our results also recover the state-of-the-art rate estimates in KL divergence, total variation and $2$-Wasserstein distance in the same setup. Finally, as we rely on the logarithmic Sobolev inequality, our framework covers a range of non-convex potentials that are first-order smooth and exhibit strong convexity outside of a compact region.

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