Learning Fast Monomial Orders for Gröbner Basis Computations

TL;DR

This work tackles the problem of selecting effective monomial orderings for Gröbner basis computations by casting it as a stationary reinforcement learning task. Using a domain-informed reward based on F4 trace statistics and a Julia-based training loop, the authors demonstrate that learned policies consistently outperform GrevLex (and often GrevLex vs GrLex baselines) across diverse zero-dimensional polynomial systems from systems biology and computer vision, achieving substantial runtime reductions. Distillation attempts reveal that the optimal strategies exploit intricate, high-dimensional geometry of the Gröbner fan and are not readily captured by simple symbolic or shallow interpretable models, underscoring the value of deep RL for symbolic computation. The work further suggests promising directions, such as graph neural networks for cross-family generalization and interpretable-by-design architectures, to bridge performance gains with interpretability and broader applicability.

Abstract

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.

Figures (10)

• Figure 1: A schematic two-dimensional slice ($w_1+w_2+w_3=1$, $w\ge0$) of the Gröbner fan for $I=\langle x+y+z,\;x^3z+x+y^2\rangle$, illustrating how regions of constant $\mathrm{in}_w(I)$ arise. The diagram is not to scale and is intended to convey the geometric structure of cones rather than exact fan boundaries; see FukJensenThomas2007.
• Figure 2: Reward improvement (relative to GrevLex) versus runtime for Gröbner basis computation on a fixed randomly generated ideal. Each point corresponds to a weighted monomial ordering with weights $(i,j,k)\in\{30,35,\ldots,70\}^3$ ($N=729$).
• Figure 3: WNT shuttle model: empirical CDF of per-instance test-time reward percent improvement relative to GrevLex, $\Delta r_{\%} = 100\,(r_{\text{agent}} - r_{\text{GrevLex}})/(|r_{\text{GrevLex}}|+\varepsilon)$ (pooled over 5 seeds; $10^5$ test ideals per seed). The dashed vertical line marks $\Delta r_{\%}=0$ (tie); values to the right indicate instances where the agent achieves higher reward than GrevLex. The median improvement is $49.6\%$ and $\Pr[\Delta r_{\%}>0]=0.916$.
• Figure 4: Mean agent reward for relative pose with unknown focal length system per training episode alongside GrevLex and GrLex baselines. Interquartile range (IQR) over seeds is shown in light blue.
• Figure 5: Mean agent reward for three-view triangulation system per training episode alongside GrevLex and GrLex baselines. Interquartile range (IQR) over seeds is shown in light blue.

Theorems & Definitions (2)

• Definition 2.1
• Definition 2.2