We propose an actor-critic algorithm for a family of complex problems arising
in algebraic statistics and discrete optimization. The core task is to produce
a sample from a finite subset of the non-negative integer lattice defined by a
high-dimensional polytope. We translate the problem into a Markov decision
process and devise an actor-critic reinforcement learning (RL) algorithm to
learn a set of good moves that can be used for sampling. We prove that the
actor-critic algorithm converges to an approximately optimal sampling policy.
To tackle complexity issues that typically arise in these sampling problems,
and to allow the RL to function at scale, our solution strategy takes three
steps: decomposing the starting point of the sample, using RL on each induced
subproblem, and reconstructing to obtain a sample in the original polytope. In
this setup, the proof of convergence applies to each subproblem in the
decomposition.
We test the method in two regimes. In statistical applications, a
high-dimensional polytope arises as the support set for the reference
distribution in a model/data fit test for a broad family of statistical models
for categorical data. We demonstrate how RL can be used for model fit testing
problems for data sets for which traditional MCMC samplers converge too slowly
due to problem size and sparsity structure. To test the robustness of the
algorithm and explore its generalization properties, we apply it to
synthetically generated data of various sizes and sparsity levels.

我们提出了一个用于代数统计和离散优化方面复杂问题的演员 - 评论算法，在高维多面体定义的有限非负整数点格子的一个子集中生成样本。我们将问题转化为马尔可夫决策过程，并设计了一种演员 - 评论增强学习算法，以学习一组可以用于采样的良好移动。我们证明了演员 - 评论算法收敛到一个近似最佳的采样策略。为了解决这些采样问题中通常出现的复杂性问题，并允许增强学习在大规模情况下发挥作用，我们的解决策略包括三个步骤：分解样本的起始点，在每个诱导的子问题上使用增强学习，并进行重构以获得原始多面体中的样本。在这个设置中，收敛性证明适用于分解中的每个子问题。我们在两个不同的情况下测试了这种方法。在统计应用中，高维多面体作为参考分布的支持集出现在广泛的分类数据统计模型的模型 / 数据拟合测试中。我们演示了如何使用增强学习解决模型拟合测试问题，对于问题规模和稀疏结构导致传统的 MCMC 采样器收敛太慢的数据集。为了测试算法的鲁棒性并探索其概括能力，我们将其应用于不同大小和稀疏水平的合成数据。

纤维采样问题的演员 - 评论算法

Actor-critic algorithms for fiber sampling problems

Gaussian mixture models are central to classical statistics, widely used in
the information sciences, and have a rich mathematical structure. We examine
their maximum likelihood estimates through the lens of algebraic statistics.
The MLE is not an algebraic function of the data, so there is no notion of ML
degree for these models. The critical points of the likelihood function are
transcendental, and there is no bound on their number, even for mixtures of two
univariate Gaussians.

本文通过代数统计学的方法分析高斯混合模型中的极大似然估计，发现 MLE 不是数据的代数函数，因此这些模型没有 ML 度的概念，似然函数的临界点是超越的，并且即使是两个单变量高斯混合模型，其临界点数量也没有界限。

高斯混合模型的最大似然估计是超越函数的

Maximum Likelihood Estimates for Gaussian Mixtures Are Transcendental

The restricted Boltzmann machine is a graphical model for binary random
variables. Based on a complete bipartite graph separating hidden and observed
variables, it is the binary analog to the factor analysis model. We study this
graphical model from the perspectives of algebraic statistics and tropical
geometry, starting with the observation that its Zariski closure is a Hadamard
power of the first secant variety of the Segre variety of projective lines. We
derive a dimension formula for the tropicalized model, and we use it to show
that the restricted Boltzmann machine is identifiable in many cases. Our
methods include coding theory and geometry of linear threshold functions.

本文从代数统计学和热带几何的角度研究了基于完全二分图的 restricted Boltzmann machine 模型及其可识别性和维数，得出了该模型的计算和解码方法和与线性门限函数的联系。