On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness
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We study sampling from a target distribution ν_* = e^-f using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function f whose tails behave like x^α for α∈ [1,2], and has β-Hölder continuous gradient, we prove that 𝒪(d^1/β+1+β/β(2/α - 1_{α≠ 1})ϵ^-1/β) steps are sufficient to reach the ϵ-neighborhood of a d-dimensional target distribution ν_* in KL-divergence. This convergence rate, in terms of ϵ dependency, is not directly influenced by the tail growth rate α of the potential function as long as its growth is at least linear, and it only relies on the order of smoothness β. One notable consequence of this result is that for potentials with Lipschitz gradient, i.e. β=1, our rate recovers the best known rate 𝒪(dϵ^-1) which was established for strongly convex potentials in terms of ϵ 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 α starts to have an effect on the established rate 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. α = 2, in the current setup. Our framework allows for finite perturbations, and any order of smoothness β∈(0,1]; consequently, our results are applicable to a wide class of non-convex potentials that are weakly smooth and exhibit at least linear tail growth.
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