Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem
This addresses bias reduction in sampling algorithms for probabilistic modeling, but it is incremental as it builds on existing Langevin dynamics methods.
The paper tackles the bias in the unadjusted Langevin algorithm (ULA) for sampling by analyzing it as an optimization problem in measure space, proposing the symmetrized Langevin algorithm (SLA) to reduce bias, and showing SLA is consistent for Gaussian targets while ULA is not.
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.