On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness

We study sampling from a target distribution $\nu_* = e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like $\|x\|^\alpha$ for ${\alpha \in [1,2]}$, and has $\beta$-H\older continuous gradient, we prove that $\widetilde{\mathcal{O}} \Big(d^{\frac{1}{\beta}+\frac{1+\beta}{\beta}(\frac{2}{\alpha}-{1}_{\{\alpha \neq 1\}})} \epsilon^{-\frac{1}{\beta}}\Big)$ steps are sufficient to reach the $\epsilon$-neighborhood of a $d$-dimensional target distribution $\nu_*$ in KL-divergence. This bound, in terms of $\epsilon$ dependency, is not directly influenced by the tail growth rate $\alpha$ of the potential function as long as its growth is at least linear, and it only relies on the order of smoothness $\beta$. One notable consequence of this result is that for potentials with Lipschitz gradient, i.e. $\beta=1$, the above rate recovers the best known rate $\widetilde{\mathcal{O}} (d\epsilon^{-1})$ which was established for strongly convex potentials in terms of $\epsilon$ dependency, but we show that the same rate is achievable for a wider class of potentials that are degenerately convex at infinity. The growth rate $\alpha$ affects the rate estimate in high dimensions where $d$ is large; furthermore, it recovers the best-known dimension dependency when the tail growth of the potential is quadratic, i.e. $\alpha = 2$, in the current setup.

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