Actor-critic algorithms for fiber sampling problems

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.

Predicting the cardinality of a reduced Gröbner basis

We use ansatz neural network models to predict key metrics of complexity for Gr\"obner bases of binomial ideals. This work illustrates why predictions with neural networks from Gr\"obner computations are not a straightforward process. Using two probabilistic models for random binomial ideals, we generate and make available a large data set that is able to capture sufficient variability in Gr\"obner complexity. We use this data to train neural networks and predict the cardinality of a reduced Gr\"obner basis and the maximum total degree of its elements. While the cardinality prediction problem is unlike classical problems tackled by machine learning, our simulations show that neural networks, providing performance statistics such as $r^2 = 0.401$, outperform naive guess or multiple regression models with $r^2 = 0.180$.

Learning a performance metric of Buchberger's algorithm

What can be (machine) learned about the complexity of Buchberger's algorithm? Given a system of polynomials, Buchberger's algorithm computes a Gr\"obner basis of the ideal these polynomials generate using an iterative procedure based on multivariate long division. The runtime of each step of the algorithm is typically dominated by a series of polynomial additions, and the total number of these additions is a hardware independent performance metric that is often used to evaluate and optimize various implementation choices. In this work we attempt to predict, using just the starting input, the number of polynomial additions that take place during one run of Buchberger's algorithm. Good predictions are useful for quickly estimating difficulty and understanding what features make Gr\"obner basis computation hard. Our features and methods could also be used for value models in the reinforcement learning approach to optimize Buchberger's algorithm introduced in [Peifer, Stillman, and Halpern-Leistner, 2020]. We show that a multiple linear regression model built from a set of easy-to-compute ideal generator statistics can predict the number of polynomial additions somewhat well, better than an uninformed model, and better than regression models built on some intuitive commutative algebra invariants that are more difficult to compute. We also train a simple recursive neural network that outperforms these linear models. Our work serves as a proof of concept, demonstrating that predicting the number of polynomial additions in Buchberger's algorithm is a feasible problem from the point of view of machine learning.