Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients (1912.04871v4)

Abstract: Discovering the underlying mathematical expressions describing a dataset is a core challenge for artificial intelligence. This is the problem of $\textit{symbolic regression}$. Despite recent advances in training neural networks to solve complex tasks, deep learning approaches to symbolic regression are underexplored. We propose a framework that leverages deep learning for symbolic regression via a simple idea: use a large model to search the space of small models. Specifically, we use a recurrent neural network to emit a distribution over tractable mathematical expressions and employ a novel risk-seeking policy gradient to train the network to generate better-fitting expressions. Our algorithm outperforms several baseline methods (including Eureqa, the gold standard for 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 constraints in situ, and a risk-seeking policy gradient formulation that optimizes for best-case performance instead of expected performance.

• The paper introduces a deep symbolic regression framework that leverages a risk-seeking policy gradient to focus on best-case rewards.
• It employs an autoregressive RNN to generate valid mathematical expressions, outperforming traditional Genetic Programming approaches on benchmark tests.
• Results indicate superior recovery rates in symbolic regression tasks, highlighting promising applications in neural architecture search and scientific discovery.

Deep Symbolic Regression: A Novel Framework Leveraging Risk-Seeking Policy Gradients

The paper presents an innovative approach to symbolic regression, a task that involves finding interpretable mathematical expressions to describe a dataset. While symbolic regression has traditionally relied on methods such as Genetic Programming (GP), the authors introduce a deep learning-based framework titled Deep Symbolic Regression (DSR) that utilizes a recurrent neural network (RNN) to generate potential solutions. A key feature of this framework is the use of a risk-seeking policy gradient to optimize the model, focusing on maximizing best-case performance rather than average performance.

Core Contributions

The primary contributions of the paper include:

1. Novel Framework for Symbolic Regression: The authors develop an autoregressive RNN that emits sequences representing possible mathematical expressions. The network is trained using a risk-seeking policy gradient, which centers on improving the best solutions found, differing from traditional policy gradients that optimize average performance.
2. Risk-Seeking Policy Gradient: This approach addresses the "expectation problem" by optimizing for high-performing samples rather than maximizing expected reward. The policy gradient reinforces successful expressions’ sequences, thus adjusting the RNN to generate higher-quality outputs over time.
3. Comparative Evaluation: The DSR framework is shown to outperform several baseline methods on the Nguyen benchmark suite, a well-recognized set of challenges in symbolic regression. Notably, its recovery rate surpasses baselines like GP and even established commercial algorithms such as Eureqa and Wolfram's approach.

Technical Approach

The RNN generates expressions by sampling each token sequentially, considering constraints to ensure expressions remain valid and meaningful. By using a tree structure to represent expressions, the method captures both the hierarchical and variable-length nature of mathematical formulas.

For evaluating potential solutions, the authors utilize a Normalized Root-Mean-Square Error (NRMSE) based fitness function modified by a squashing transformation. This ensures a bounded and efficient optimization objective.

Implications and Future Directions

The adoption of a risk-seeking policy in a reinforcement learning framework aligns DSR with domains requiring high-performance solutions in combinatorially large search spaces. Potential applications beyond symbolic regression include neural architecture search and search tasks in molecular biology or physics where interpretability is crucial.

A fascinating aspect is the potential for integrating the DSR framework with existing methods such as GP, which could lead to hybrid techniques leveraging the strengths of both evolutionary algorithms and neural networks.

The framework's adaptability to handle hierarchical data structures opens avenues for novel applications, such as optimizing functions in control policies or chemistry for drug discovery. Moreover, the risk-seeking policy gradient may inspire further exploration in reinforcement learning, particularly for tasks emphasizing extreme performance outcomes.

Overall, this work pushes the boundaries of symbolic regression by combining deep learning with reinforcement learning in a novel way, offering both theoretical insights and practical tools for discovering interpretable models in complex datasets.