用连续时间马尔可夫链设计高效离散采样算法，降低重要性权重方差。
LEAPS: A discrete neural sampler via locally equivariant networks
- 通过局部等变函数构建速率矩阵，实现对离散分布的高效采样。
- 在统计物理问题中验证，相比传统方法显著降低采样方差。
- 适合需要高精度采样的机器学习与物理模拟场景。
我们提出LEAPS，一种通过学习连续时间马尔可夫链（CTMC）的速率矩阵来从归一化未知的离散分布中采样的算法。LEAPS可视为退火重要性采样与顺序蒙特卡洛方法的连续时间形式，其通过引入CTMC使重要性权重的方差得以控制。为计算这些权重，我们引入了基于路径测度的柯尔莫哥洛夫导数。由于标准神经网络参数化下该计算不可行，我们提出一种称为“局部等变”的速率矩阵紧凑表示法，并设计了具备局部等变性的多层感知机、注意力层和卷积网络，提供保持该性质的深层网络构造方法。该特性使得我们能够设计出一种可扩展的训练算法，使关联于CTMC的重要性权重方差最小化。我们在统计物理问题中验证了LEAPS的有效性。
原文摘要 · Abstract (English)
We propose "LEAPS", an algorithm to sample from discrete distributions known up to normalization by learning a rate matrix of a continuous-time Markov chain (CTMC). LEAPS can be seen as a continuous-time formulation of annealed importance sampling and sequential Monte Carlo methods, extended so that the variance of the importance weights is offset by the inclusion of the CTMC. To derive these importance weights, we introduce a set of Radon-Nikodym derivatives of CTMCs over their path measures. Because the computation of these weights is intractable with standard neural network parameterizations of rate matrices, we devise a new compact representation for rate matrices via what we call "locally equivariant" functions. To parameterize them, we introduce a family of locally equivariant multilayer perceptrons, attention layers, and convolutional networks, and provide an approach to make deep networks that preserve the local equivariance. This property allows us to propose a scalable training algorithm for the rate matrix such that the variance of the importance weights associated to the CTMC are minimal. We demonstrate the efficacy of LEAPS on problems in statistical physics.
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