Title: Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem,
Abstract: We study sampling as optimization in the space of measures. We focus on
gradient flow-based optimization with the Langevin dynamics as a case study. We
investigate the source of the bias of the unadjusted Langevin algorithm (ULA)
in discrete time, and consider how to remove or reduce the bias. We point out
the difficulty is that the heat flow is exactly solvable, but neither its
forward nor backward method is implementable in general, except for Gaussian
data. We propose the symmetrized Langevin algorithm (SLA), which should have a
smaller bias than ULA, at the price of implementing a proximal gradient step in
space. We show SLA is in fact consistent for Gaussian target measure, whereas
ULA is not. We also illustrate various algorithms explicitly for Gaussian
target measure, including gradient descent, proximal gradient, and
Forward-Backward, and show they are all consistent.