We propose a class of discrete state sampling algorithms based on Nesterov's accelerated gradient method, which extends the classical Metropolis-Hastings (MH) algorithm. The evolution of the discrete states probability distribution governed by MH can be interpreted as a gradient descent direction of the Kullback--Leibler (KL) divergence, via a mobility function and a score function. Specifically, this gradient is defined on a probability simplex equipped with a discrete Wasserstein-2 metric with a mobility function. This motivates us to study a momentum-based acceleration framework using damped Hamiltonian flows on the simplex set, whose stationary distribution matches the discrete target distribution. Furthermore, we design an interacting particle system to approximate the proposed accelerated sampling dynamics. The extension of the algorithm with a general choice of potentials and mobilities is also discussed. In particular, we choose the accelerated gradient flow of the relative Fisher information, demonstrating the advantages of the algorithm in estimating discrete score functions without requiring the normalizing constant and keeping positive probabilities. Numerical examples, including sampling on a Gaussian mixture supported on lattices or a distribution on a hypercube, demonstrate the effectiveness of the proposed discrete-state sampling algorithm.

翻译：我们提出了一类基于Nesterov加速梯度方法的离散状态采样算法，该算法扩展了经典的Metropolis-Hastings（MH）算法。通过引入迁移函数和得分函数，MH算法所控制的离散状态概率分布的演化可被解释为Kullback-Leibler（KL）散度的梯度下降方向。具体而言，该梯度定义在配备具有迁移函数的离散Wasserstein-2度量的概率单纯形上。这促使我们研究在单纯形集上使用阻尼哈密顿流的动量加速框架，其平稳分布与离散目标分布相匹配。此外，我们设计了一个相互作用的粒子系统来逼近所提出的加速采样动力学。文中还讨论了具有一般势函数和迁移函数选择的算法扩展。特别地，我们选择了相对Fisher信息的加速梯度流，展示了该算法在估计离散得分函数时无需归一化常数且能保持概率正性的优势。数值实验（包括在格点支撑的高斯混合分布或超立方体上的分布进行采样）验证了所提出的离散状态采样算法的有效性。