Iterative Exact Discrete Guidance for Energy-Based Sampling

2026-09-29

AI Summary

This study addresses the challenge of sampling from multimodal target distributions over large discrete spaces by proposing the Iterative Energy-Guided Diffusion (IEDG) framework. IEDG achieves precise discrete guided sampling through global Boltzmann tilting combined with iterative local corrections, and introduces a novel trajectory-level exact guidance mechanism that adaptively controls the thermodynamic geometric step size of the annealing schedule via the relative effective sample size. Theoretically, this work establishes an error analysis framework based on Rényi-2 divergence and total variation distance. Empirically, IEDG significantly outperforms existing neural baselines on Ising, Potts, and Max-Cut benchmarks, substantially reducing single-step errors while improving distribution coverage.

Abstract

Sampling from unnormalized distributions over large discrete state spaces becomes difficult when a multimodal target is far from a tractable reference. We introduce Iterative Exact Discrete Guidance (IEDG), a population-exact, trajectory-wise guidance framework for unnormalized discrete targets. Rather than learn the full reference-to-target correction in one step, IEDG introduces a global Boltzmann tilt along an annealing trajectory. Each stage learns a stage-local posterior correction for an incremental Boltzmann tilt of the current source, while the resulting corrections are accumulated relative to a fixed analytic posterior. At the population optimum, exact stage posteriors recover the correct reverse dynamics, whose exact simulation reproduces the target distribution. IEDG chooses stage increments by relative effective sample size (rESS), which controls Rényi-2 displacement and locally adapts the step size to the thermodynamic geometry of the annealing path. Our stagewise total-variation analysis shows that limited overlap amplifies Bregman fitting error by $1/\sqrt{\mathrm{rESS}}$, while posterior, simulation, and truncation errors enter additively. IEDG improves all distribution-level errors over the neural baselines on ordered, exactly enumerated Ising $4\times4$, while substantially reducing one-shot errors on Ising/Potts $16\times16$ across thermodynamic regimes and attaining the best neural-sampler result on several reported local-statistic and phase-coverage metrics. On Max-Cut, its best-of-512 and average-sample ratios exceed all the baselines. Code and artifacts are available at https://github.com/StillFantasy123/iterative-exact-discrete-guidance.

Problem

• discrete sampling
• unnormalized distributions
• energy-based models
• large state spaces
• multimodal targets

Innovation

• Iterative Exact Discrete Guidance
• Energy-Based Sampling
• Annealing Trajectory
• Relative Effective Sample Size
• Discrete State Spaces