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.

翻译：我们发展了一套离散最优传输框架，用于分析有限状态空间上的模拟退火算法。基于Maas（J. Funct. Anal., 2011）引入的离散Wasserstein度量，我们定义了广义的离散Wasserstein-2距离以及图上概率测度路径的关联概念——离散作用量。利用这些工具，我们建立了模拟退火的非渐近收敛保证：算法输出与目标分布之间的KL散度受退火路径的离散作用量控制。这可视为郭、陶和陈（ICLR 2025）在连续空间中基于作用量分析的退火Langevin动力学的离散对应。作为应用，我们分析了统计物理学中两个基础模型的模拟退火。对于平均场伊辛模型，我们证明退火单点Glauber动力学在任意逆温度β≥0下，需O(n^5β^2/ε)步达到KL散度ε误差。对于平均场q态Potts模型，我们证明退火(q-1)块Glauber动力学对所有β≥β_s=q/2（即无序相完全失去稳定性的区域）可在poly(n, β, 1/ε)步内达到ε误差。在这两种情形中，关键的技术贡献是通过利用模型对称性将分析简化为低维投影链，从而得到离散作用量的多项式上界。