Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentials
This work addresses sampling challenges in high-dimensional statistics for researchers, though it is incremental as it builds on prior methods by relaxing conditions.
The paper tackles the problem of sampling from mixtures of weakly smooth potentials using an unadjusted Langevin algorithm, establishing convergence in KL divergence with polynomial dimension dependence and relaxing previous convexity assumptions.
Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, it seems to be a considerable restriction when the potentials are often required to be smooth (gradient Lipschitz). This paper studies the problem of sampling through Euler discretization, where the potential function is assumed to be a mixture of weakly smooth distributions and satisfies weakly dissipative. We establish 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. We relax the degenerated convex at infinity conditions of \citet{erdogdu2020convergence} and prove convergence guarantees under Poincaré inequality or non-strongly convex outside the ball. In addition, we also provide convergence in $L_β$-Wasserstein metric for the smoothing potential.