# Vertical Symbolic Regression

CoRR（2023）

Abstract

Automating scientific discovery has been a grand goal of Artificial
Intelligence (AI) and will bring tremendous societal impact. Learning symbolic
expressions from experimental data is a vital step in AI-driven scientific
discovery. Despite exciting progress, most endeavors have focused on the
horizontal discovery paths, i.e., they directly search for the best expression
in the full hypothesis space involving all the independent variables.
Horizontal paths are challenging due to the exponentially large hypothesis
space involving all the independent variables. We propose Vertical Symbolic
Regression (VSR) to expedite symbolic regression. The VSR starts by fitting
simple expressions involving a few independent variables under controlled
experiments where the remaining variables are held constant. It then extends
the expressions learned in previous rounds by adding new independent variables
and using new control variable experiments allowing these variables to vary.
The first few steps in vertical discovery are significantly cheaper than the
horizontal path, as their search is in reduced hypothesis spaces involving a
small set of variables. As a consequence, vertical discovery has the potential
to supercharge state-of-the-art symbolic regression approaches in handling
complex equations with many contributing factors. Theoretically, we show that
the search space of VSR can be exponentially smaller than that of horizontal
approaches when learning a class of expressions. Experimentally, VSR
outperforms several baselines in learning symbolic expressions involving many
independent variables.

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