Exhaustive Symbolic Regression by Deaglan J. Bartlett et al. on Tuesday 22 November

Symbolic Regression (SR) algorithms learn analytic expressions which both

accurately fit data and, unlike traditional machine-learning approaches, are

highly interpretable. Conventional SR suffers from two fundamental issues which

we address in this work. First, since the number of possible equations grows

exponentially with complexity, typical SR methods search the space

stochastically and hence do not necessarily find the best function. In many

cases, the target problems of SR are sufficiently simple that a brute-force

approach is not only feasible, but desirable. Second, the criteria used to

select the equation which optimally balances accuracy with simplicity have been

variable and poorly motivated. To address these issues we introduce a new

method for SR -- Exhaustive Symbolic Regression (ESR) -- which systematically

and efficiently considers all possible equations and is therefore guaranteed to

find not only the true optimum but also a complete function ranking. Utilising

the minimum description length principle, we introduce a principled method for

combining these preferences into a single objective statistic. 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 considered) that fit

the data more economically than the Friedmann equation. These low-redshift data

therefore do not necessarily prefer a $\Lambda$CDM expansion history, and

traditional SR algorithms that return only the Pareto-front, even if they found

this successfully, would not locate $\Lambda$CDM. We make our code and full

equation sets publicly available.

arXiv: http://arxiv.org/abs/http://arxiv.org/abs/2211.11461v1