Non-convex sampling for a mixture of locally smooth potentials
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The purpose of this paper is to examine the sampling problem through Euler discretization, where the potential function is assumed to be a mixture of locally smooth distributions and weakly dissipative. We introduce α_G-mixture locally smooth and α_H-mixture locally Hessian smooth, which are novel and typically satisfied with a mixture of distributions. Under our conditions, we prove the convergence in Kullback-Leibler (KL) divergence with the number of iterations to reach ϵ-neighborhood of a target distribution in only polynomial dependence on the dimension. The convergence rate is improved when the potential is 1-smooth and α_H-mixture locally Hessian smooth. Our result for the non-strongly convex outside the ball of radius R is obtained by convexifying the non-convex domains. In addition, we provide some nice theoretical properties of p-generalized Gaussian smoothing and prove the convergence in the L_β-Wasserstein distance for stochastic gradients in a general setting.
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