We study the problem of sampling from a distribution where the negative logarithm of the target density is -smooth everywhere and -strongly convex outside a ball of radius , 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 of the target distribution in -Wasserstein distance. For the first-order method (overdamped Langevin MCMC), the time complexity is , where is the dimension of the underlying space. For the second-order method (underdamped Langevin MCMC), the time complexity is for some explicit positive constant . Surprisingly, the convergence rate is only polynomial in the dimension and the target accuracy . It is however exponential in the problem parameter , which is a measure of non-logconcavity of the target distribution.
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