Sharp Convergence Rates for Langevin Dynamics in the Nonconvex Setting
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We study the problem of sampling from a distribution where the negative logarithm of the target density is L-smooth everywhere and m-strongly convex outside a ball of radius R, but potentially non-convex inside this ball. We study both overdamped and underdamped Langevin MCMC and prove upper bounds on the time required to obtain a sample from a distribution that is within ϵ of the target distribution in 1-Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the time complexity is Õ(e^cLR^2d/ϵ^2), where d is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the time complexity is Õ(e^cLR^2√(d)/ϵ) for some explicit positive constant c. Surprisingly, the convergence rate is only polynomial in the dimension d and the target accuracy ϵ. It is however exponential in the problem parameter LR^2, which is a measure of non-logconcavity of the target distribution.
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