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We study sampling from a target distribution ${\nu_* = e^{-f}}$ using theunadjusted 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 ${\widetilde{\mathcal{O}}\Big(d^{\frac{1}{\beta}+\frac{1+\beta}{\beta}(\frac{2}{\alpha} -\boldsymbol{1}_{\{\alpha \neq 1\}})} \epsilon^{-\frac{1}{\beta}}\Big)}$ stepsare sufficient to reach the $\epsilon $-neighborhood of a $d$-dimensionaltarget distribution $\nu_*$ in KL-divergence. This convergence rate, in termsof $\epsilon$ 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 consequenceof this result is that for potentials with Lipschitz gradient, i.e. $\beta=1$,our rate recovers the best known rate${\widetilde{\mathcal{O}}(d\epsilon^{-1})}$ which was established for stronglyconvex potentials in terms of $\epsilon$ dependency, but we show that the samerate is achievable for a wider class of potentials that are degenerately convexat infinity. The growth rate $\alpha$ starts to have an effect on theestablished rate in high dimensions where $d$ is large; furthermore, itrecovers the best-known dimension dependency when the tail growth of thepotential is quadratic, i.e. ${\alpha = 2}$, in the current setup. Ourframework allows for finite perturbations, and any order of smoothness${\beta\in(0,1]}$; consequently, our results are applicable to a wide class ofnon-convex potentials that are weakly smooth and exhibit at least linear tailgrowth.

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