Learning Fast Monomial Orders for Gröbner Basis Computations

The efficiency of Gröbner basis computation, the standard engine for solving systems of polynomial equations, depends on the choice of monomial ordering. Despite a near-continuum of possible monomial orders, most implementations rely on static heuristics such as GrevLex, guided primarily by expert intuition. We address this gap by casting the selection of monomial orderings as a reinforcement learning problem over the space of admissible orderings. Our approach leverages domain-informed reward signals that accurately reflect the computational cost of Gröbner basis computations and admits efficient Monte Carlo estimation. Experiments on benchmark problems from systems biology and computer vision show that the resulting learned policies consistently outperform standard heuristics, yielding substantial reductions in computational cost. Moreover, we find that these policies resist distillation into simple interpretable models, providing empirical evidence that deep reinforcement learning allows the agents to exploit non-linear geometric structure beyond the scope of traditional heuristics.

💡 Research Summary

This paper tackles a long‑standing bottleneck in computational algebra: the choice of monomial ordering for Gröbner basis computation. While the space of admissible orderings is essentially continuous (the Gröbner fan), most computer algebra systems restrict themselves to a handful of static heuristics such as Lex, GrLex, and GrevLex. The authors cast the selection of an ordering as a reinforcement‑learning (RL) problem, thereby allowing an algorithm to discover orderings that are tailored to the structure of each polynomial system.

The core of the approach is a simple yet expressive action space. An ordering is represented by a weight vector w∈ℝⁿ₊ with ∑w_i=1 (the simplex). In practice the RL agent outputs a point on the simplex; this point is scaled by 10³ and rounded to integers because the Julia package Groebner.jl, which the authors use for the underlying algebra, only accepts integer weight vectors. The environment consists of thousands of randomly generated zero‑dimensional polynomial systems (i.e., systems with finitely many solutions). For each system the agent selects a weight vector, the Gröbner basis is computed using the F4 algorithm, and a set of trace statistics is recorded.

The reward function is designed to be a cheap proxy for actual runtime. It aggregates, over all F4 iterations, the product of the number of columns in the elimination matrix (n_M), the number of S‑pairs processed (n_P), and the logarithm of the pair degree (ln d_P):

 R = – Σ_{i=1}^{t} n_M(i) · n_P(i) · ln(d_P(i))

The authors validate this surrogate by sweeping 729 different weighted orderings on a fixed random ideal and showing a strong monotonic relationship between the reward and measured wall‑clock time (Pearson r = –0.949, Spearman ρ = –0.937). By anchoring the reward to GrevLex (i.e., agents receive positive reward only when they beat GrevLex), the learning signal is further stabilized.

Training is performed with Group Relative Policy Optimization (GRPO), a policy‑gradient method that has been shown to work well when the reward aligns closely with the policy’s direction. The agents are trained on a diverse collection of benchmark problems drawn from systems biology (e.g., biochemical network steady‑state equations) and computer vision (e.g., minimal solvers for multi‑view geometry). Across these domains the learned policies consistently outperform GrevLex, achieving average reductions of 30 %–45 % in the surrogate cost and up to 70 % in the most challenging instances.

A key scientific contribution is the empirical demonstration that the learned policies exploit subtle geometric features of the Gröbner fan that are not captured by simple interpretable models. The authors attempt to distill the policies into symbolic regression formulas and shallow soft decision trees (depth ≤ 3). While the distilled models can approximate the behavior on a subset of problems, they fail to reproduce the full performance gain, indicating that the policies rely on non‑linear interactions among the weight components that are difficult to express with low‑complexity models.

Implementation-wise, the authors note that mainstream Python algebra libraries do not expose the full weighted‑ordering search space, prompting a switch to Julia’s Groebner.jl. They built the RL environment from scratch, released the code and data publicly, and provide detailed documentation to facilitate reproducibility.

The paper concludes with several promising directions: (1) joint optimization of monomial ordering and Buchberger pair‑selection strategies, (2) richer action spaces (e.g., hierarchical weight scaling, adaptive simplex constraints), (3) development of more expressive yet interpretable surrogate models (e.g., neural‑symbolic hybrids) to explain the learned policies, and (4) extension of the RL framework to other symbolic‑computation tasks such as cylindrical algebraic decomposition or elimination theory.

In summary, this work demonstrates that reinforcement learning can move beyond heuristic tuning in computational algebra, automatically discovering monomial orderings that significantly reduce Gröbner basis computation time. It bridges a gap between deep learning and symbolic mathematics, offering both practical speed‑ups for real‑world polynomial systems and new insights into the structure of the Gröbner fan.