SIAM Journal on Computing, Volume 50, Issue 2, Page 350-381, January 2021.
In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop, and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or antiferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, and change of functions). The algorithms for both these models are efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new “equilibrium" property that is shown to be satisfied by our algorithm at each step and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

中文翻译：

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从图形模型动态采样

SIAM Journal on Computing，第 50 卷，第 2 期，第 350-381 页，2021 年 1 月。
在本文中，我们研究了当模型本身随时间动态变化时从图形模型中采样的问题。这个问题源于机器学习、计算机视觉、统计物理学和理论计算机科学中的各种推理、学习和采样设置。虽然从静态图形模型中采样的问题受到了相当多的关注，但其动态变体的理论工作却在很大程度上缺乏。本文的主要贡献是一种算法，该算法可以从离散随机变量的广泛图形模型中动态采样。我们的算法与拉斯维加斯并行：它知道何时停止，并从精确分布中输出样本。我们还提供了足够的条件，在该条件下，该算法在与一般图形模型以及经过充分研究的特定自旋系统的更新大小成比例的时间运行。特别是，对于 Ising 模型（铁磁或反铁磁）和硬核模型，我们获得了第一个可以处理边缘和顶点更新（添加、删除和更改函数）的动态采样算法。这两个模型的算法在接近各自唯一性的范围内是有效的，超过这个范围，即使对于静态和近似采样，也没有已知的局部算法或问题本身是棘手的。我们的动态采样算法依赖于局部重采样算法和新的“平衡” 我们的算法在每一步都显示出满足的性质，并使我们能够证明其正确性。这种均衡特性足够稳健，可以保证我们算法的正确性，帮助我们提高特定模型快速收敛的界限，并且应该具有独立的兴趣。