Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients

Discovering the underlying mathematical expressions describing a dataset is a core challenge for artificial intelligence. This is the problem of $\textit{symbolic}$ $\textit{regression.}$ Despite recent advances in training neural networks to solve complex tasks, deep learning approaches to symbolic regression are underexplored. We propose a framework that combines deep learning with symbolic regression via a simple idea: use a large model to search the space of small models. More specifically, we use a recurrent neural network to emit a distribution over tractable mathematical expressions, and employ reinforcement learning to train the network to generate better-fitting expressions. Our algorithm significantly outperforms standard genetic programming-based symbolic regression in its ability to exactly recover symbolic expressions on a series of benchmark problems, both with and without added noise. More broadly, our contributions include a framework that can be applied to optimize hierarchical, variable-length objects under a black-box performance metric, with the ability to incorporate a priori constraints in situ, and a risk-seeking policy gradient formulation that optimizes for best-case performance instead of expected performance.