Discrete Optimal Transport: Rapid Convergence of Simulated Annealing Algorithms

2026-05-07

AI Summary

This work establishes the first non-asymptotic convergence guarantee for simulated annealing in discrete state spaces. By constructing an optimal transport framework based on the discrete Wasserstein-2 metric, the authors introduce a discrete action functional to characterize annealing trajectories and control the Kullback–Leibler (KL) divergence between the output and target distributions. They innovatively extend action-based analysis from continuous spaces to discrete graph structures, leveraging model symmetries to project high-dimensional chains onto low-dimensional subspaces and thereby derive polynomial upper bounds on the action. Applying this framework, they prove that for the mean-field Ising model, Glauber dynamics achieves ε-KL accuracy within O(n⁵β²/ε) steps at any temperature; for the q-state Potts model with β ≥ q/2, (q−1)-block Glauber dynamics converges in poly(n, β, 1/ε) steps.

Abstract

We develop a discrete optimal transport framework for analyzing simulated annealing algorithms on finite state spaces. Building on the discrete Wasserstein metric introduced by Maas (J. Funct. Anal., 2011), we define a generalized discrete Wasserstein-2 distance and the associated notion of \emph{discrete action} for paths of probability measures on graphs. Using these tools, we establish non-asymptotic convergence guarantees for simulated annealing: the KL divergence between the algorithm's output and the target distribution is controlled by the discrete action of the annealing path. This can be viewed as the discrete counterpart of the action-based analysis of annealed Langevin dynamics in continuous spaces by Guo, Tao, and Chen (ICLR 2025). As applications, we analyze simulated annealing for two fundamental models in statistical physics. For the \emph{mean-field Ising model}, we show that annealed single-site Glauber dynamics achieves $\varepsilon$ error in KL divergence in $O(n^5β^2/\varepsilon)$ steps at \emph{any} inverse temperature $β\ge 0$. For the \emph{mean-field $q$-state Potts model}, we show that annealed $(q-1)$-block Glauber dynamics achieves $\varepsilon$ error in $\mathrm{poly}(n, β, 1/\varepsilon)$ steps for all $β\ge β_{\mathsf{s}}=q/2$, the regime where the disordered phase has completely lost stability. In both cases, the key technical contribution is a polynomial upper bound on the discrete action, obtained by exploiting the symmetry of the model to reduce the analysis to a low-dimensional projected chain.

Problem

• discrete optimal transport
• simulated annealing
• non-asymptotic convergence
• KL divergence
• discrete Wasserstein distance

Innovation

• discrete optimal transport
• simulated annealing
• discrete Wasserstein distance
• non-asymptotic convergence
• discrete action