Exhaustive Symbolic Regression

Abstract

Symbolic Regression (SR) algorithms attempt to learn analytic expressions which fit data accurately and in a highly interpretable manner. Conventional SR suffers from two fundamental issues which we address here. First, these methods search the space stochastically (typically using genetic programming) and hence do not necessarily find the best function. Second, the criteria used to select the equation optimally balancing accuracy with simplicity have been variable and subjective. To address these issues we introduce Exhaustive Symbolic Regression (ESR), which systematically and efficiently considers all possible equations -- made with a given basis set of operators and up to a specified maximum complexity -- and is therefore guaranteed to find the true optimum (if parameters are perfectly optimised) and a complete function ranking subject to these constraints. We implement the minimum description length principle as a rigorous method for combining these preferences into a single objective. To illustrate the power of ESR we apply it to a catalogue of cosmic chronometers and the Pantheon+ sample of supernovae to learn the Hubble rate as a function of redshift, finding $\sim$40 functions (out of 5.2 million trial functions) that fit the data more economically than the Friedmann equation. These low-redshift data therefore do not uniquely prefer the expansion history of the standard model of cosmology. We make our code and full equation sets publicly available.

15 pages, 7 figures, 2 tables. Accepted for publication in the IEEE Transactions on Evolutionary Computation

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• The Inefficiency of Genetic Programming for Symbolic Regression
• cp3-bench: A tool for benchmarking symbolic regression algorithms tested with cosmology
• Improving Genetic Programming for Symbolic Regression with Equality Graphs
• Call for Action: towards the next generation of symbolic regression benchmark
• Machine Learning-Based Analytical Expressions for Gray-Body Factors and Application to Primordial Black Holes
• Constraining dark matter halo profiles with symbolic regression
• Statistical Patterns in the Equations of Physics and the Emergence of a Meta-Law of Nature
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