Bayesian Symbolic Regression with Entropic Reinforcement Learning

2026-08-10

AI Summary

This work addresses the challenge of quantifying model structural uncertainty in symbolic regression, a task where conventional approaches typically return only a single optimal expression and overlook the diversity of the solution space. The authors propose ERRLESS, a novel method that introduces maximum-entropy reinforcement learning into Bayesian symbolic regression for the first time. ERRLESS employs a neural policy network to sequentially construct abstract syntax trees, enabling sampling of algebraic expressions from the posterior distribution. The approach naturally incorporates prior constraints, quantifies structural uncertainty, and yields concise, interpretable expressions. On the Feynman benchmark suite, ERRLESS achieves significantly higher R² scores for posterior predictive means compared to sequential Monte Carlo baselines, while maintaining short and computationally efficient expressions.

Abstract

Symbolic regression is the problem of finding an algebraic expression describing a stochastic dependence of a target variable on a set of inputs. Unlike forms of regression that fit parameters assuming a fixed model structure, symbolic regression is a search problem over the space of expressions, represented, for example, as abstract syntax trees using a library of operators. Symbolic regression is typically used in settings with limited, noisy data in the natural sciences. However, searching for a single best-fitting expression fails to capture the epistemic uncertainty about the expression, which motivates a Bayesian perspective that enables uncertainty quantification and specification of natural priors to constrain the search space. In this work, we propose ERRLESS (Entropy-Regularized Reinforcement Learning for Expression Structure Sampling), a scalable approach for sampling from the posterior distribution over expressions given data using maximum-entropy reinforcement learning. ERRLESS learns a neural policy that constructs expressions sequentially by building up their abstract syntax trees. At convergence, the policy samples expressions from the posterior. At test time, expressions can be sampled by rollouts of this policy. We demonstrate that ERRLESS achieves competitive results on the Feynman benchmark while producing short and interpretable expressions. Additionally, we demonstrate that the mean of the posterior predictive approximated by ERRLESS achieves a high coefficient of determination ($R^2$) compared to an SMC baseline, highlighting the benefits of the Bayesian perspective in symbolic regression.

Problem

• symbolic regression
• Bayesian inference
• epistemic uncertainty
• posterior sampling
• expression discovery

Innovation

• Bayesian symbolic regression
• entropy-regularized reinforcement learning
• posterior sampling
• abstract syntax trees
• uncertainty quantification