Sampling from Energy distributions with Target Concrete Score Identity
Sergei Kholkin, Francisco Vargas, Alexander Korotin
TL;DR
The paper introduces the Target Concrete Score Identity Sampler (TCSIS) for drawing samples from unnormalized discrete densities by learning reverse CTMC dynamics under a forward uniform noising process. Central to TCSIS is the Target Concrete Score Identity, which expresses the concrete score as a ratio of forward-diffusion expectations, enabling Monte Carlo estimation without sampling from the target or computing the partition function. Two practical implementations are proposed: Self-Normalized TCSIS, which directly learns the concrete score via SNIS losses, and Unbiased TCSIS, which learns marginal densities and constructs the score with unbiased losses. Experiments on Ising lattice models demonstrate competitive performance against established baselines, with SNIS showing strong accuracy and Unbiased TCSIS offering favorable trade-offs between bias and variance, highlighting the method's potential for scalable sampling from unnormalized energies in discrete spaces.
Abstract
We introduce the Target Concrete Score Identity Sampler (TCSIS), a method for sampling from unnormalized densities on discrete state spaces by learning the reverse dynamics of a Continuous-Time Markov Chain (CTMC). Our approach builds on a forward in time CTMC with a uniform noising kernel and relies on the proposed Target Concrete Score Identity, which relates the concrete score, the ratio of marginal probabilities of two states, to a ratio of expectations of Boltzmann factors under the forward uniform diffusion kernel. This formulation enables Monte Carlo estimation of the concrete score without requiring samples from the target distribution or computation of the partition function. We approximate the concrete score with a neural network and propose two algorithms: Self-Normalized TCSIS and Unbiased TCSIS. Finally, we demonstrate the effectiveness of TCSIS on problems from statistical physics.