Discrete Langevin Sampler via Wasserstein Gradient Flow

Understand

It is known that gradient-based MCMC samplers for continuous spaces, such as Langevin Monte Carlo (LMC), can be derived as particle versions of a gradient flow that minimizes KL divergence on a Wasserstein manifold.

• The superior efficiency of such samplers has motivated several recent attempts to generalize LMC to discrete spaces.
• However, a fully principled extension of Langevin dynamics to discrete spaces has yet to be achieved, due to the lack of well-defined gradients in the sample space.
• In this work, we show how the Wasserstein gradient flow can be generalized naturally to discrete spaces.