End-to-end symbolic regression with transformers (2204.10532v1)

Abstract: Symbolic regression, the task of predicting the mathematical expression of a function from the observation of its values, is a difficult task which usually involves a two-step procedure: predicting the "skeleton" of the expression up to the choice of numerical constants, then fitting the constants by optimizing a non-convex loss function. The dominant approach is genetic programming, which evolves candidates by iterating this subroutine a large number of times. Neural networks have recently been tasked to predict the correct skeleton in a single try, but remain much less powerful. In this paper, we challenge this two-step procedure, and task a Transformer to directly predict the full mathematical expression, constants included. One can subsequently refine the predicted constants by feeding them to the non-convex optimizer as an informed initialization. We present ablations to show that this end-to-end approach yields better results, sometimes even without the refinement step. We evaluate our model on problems from the SRBench benchmark and show that our model approaches the performance of state-of-the-art genetic programming with several orders of magnitude faster inference.

• The paper presents a novel Transformer model that jointly predicts both mathematical structure and constants in a single pass.
• The paper’s methodology integrates deep learning with non-convex optimization to improve accuracy and computational efficiency.
• The paper achieves near state-of-the-art performance with significantly faster inference compared to traditional genetic programming methods.

End-to-End Symbolic Regression with Transformers

The paper "End-to-End Symbolic Regression with Transformers" presents a novel approach for performing symbolic regression (SR) using Transformers, a deep learning architecture that has recently achieved significant successes in various fields, particularly in natural language processing. The work aims to improve the efficiency and accuracy of symbolic regression tasks by employing a direct end-to-end model that predicts both the structure and the constants of mathematical expressions, as opposed to the traditional two-step procedure involving skeleton prediction followed by constant fitting.

Overview of Symbolic Regression and Current Challenges

Symbolic regression involves inferring the mathematical expression of a function from observed data points. The typical method involves predicting an expression's "skeleton" and then fitting numerical constants using optimization techniques, usually non-convex optimization. The dominant SR methodologies rely on genetic programming (GP), which iteratively refines candidate solutions and problems. However, GP has limitations, including high computational costs and an inability to leverage past learning since every new problem is approached from scratch.

Prior approaches have utilized neural networks for SR, focusing on predicting the function skeleton on a single pass, but these approaches have struggled to match GP in terms of accuracy.

The Transformer-Based Approach

This paper proposes utilizing Transformers to perform symbolic regression in an end-to-end manner. The paper leverages the capability of Transformers to model sequences by simultaneously predicting the full mathematical expression, including both the structure and numerical constants. The model is further refined using a non-convex optimizer (e.g., BFGS), with the constants predicted by the Transformer acting as a well-informed initialization. This hybrid approach marries the strength of deep learning with the precision of traditional optimization techniques.

A critical novelty of the model is its vocabulary, which combines symbolic tokens for operators and variables with numeric tokens for constants. This approach demonstrates that such a combined vocabulary can effectively tackle SR tasks traditionally seen as challenging for symbolic and numeric reasons.

Empirical Results

The model is trained on synthetic datasets and evaluated using the SRBench benchmark. It delivers results approaching those of state-of-the-art GP techniques but with considerably faster inference times, achieving performance magnitudes several orders faster. Significant accuracy gains are observed, particularly in generating less complex expressions and displaying strong robustness against noise and extrapolation capabilities.

Furthermore, various ablations show that as complexity factors like the number of unary operators, binary operators, or input dimensions increase, the ablation reveals robustness in performance due to the end-to-end prediction strategy. The refinement process further enhances predictions, especially for expressions involving higher-dimensional input spaces or noisy datasets.

Implications and Future Directions

The promising results from this paper suggest that deep learning models, specifically Transformers, can be effectively employed for symbolic regression tasks. This opens doors to applications in areas requiring rapid computation such as reinforcement learning and physics simulations, thus expanding the potential reach of SR methodologies beyond traditional genetic programming.

Future research can focus on scaling this methodology to handle even larger input dimensions and exploring the applicability of this approach in truly high-dimensional contexts. Moreover, further improvements may be realized by optimizing the data generation methods and refining the neural architecture to better capture the intricacies of symbolic regressions.

In summary, this work represents a significant contribution to the field of symbolic regression, offering a robust and efficient alternative to traditional GP and highlighting the transformative potential of neural architectures in solving complex mathematical tasks.