Overview
This work presents a learning-theoretic analysis of symbolic regression (SR) models, specifically those utilizing genetic programming (GP) and represented as expression trees. The primary objective is to advance the theoretical understanding of generalization in GP-based SR, particularly why these methods successfully generalize beyond their training data. The analysis introduces a generalization bound under specified constraints, offering a structured view of the factors influencing generalization performance.
Research Context
Symbolic regression with genetic programming aims to directly derive interpretable mathematical expressions from data. Despite its observed empirical effectiveness, the theoretical basis for its generalization capabilities has remained an area with limited understanding. This research addresses this gap by providing a theoretical framework to explain the generalization properties of GP-based SR, connecting practical design choices to explicit components within the generalization bound.
Approach
The research methodology involves a learning-theoretic analysis of SR models. It focuses on models structured as expression trees. A generalization bound is derived for GP-style SR, contingent on constraints related to tree size, tree depth, and the manageability of learnable constants within the expressions. The derivation of this bound aims to dissect the generalization gap into distinct, interpretable components.
Findings
- The derived generalization bound for GP-style SR decomposes the generalization gap into two primary components:
- Structure-selection term: This component reflects the combinatorial complexity associated with selecting an appropriate expression-tree structure.
- Constant-fitting term: This component captures the complexity involved in optimizing numerical constants within a fixed, predetermined expression structure.
- This decomposition offers a theoretical perspective on several commonly employed practices within genetic programming:
- Parsimony pressure: The analysis suggests how structural restrictions, such as those encouraged by parsimony pressure, contribute to reducing the growth of the hypothesis class.
- Depth limits: Imposing limits on tree depth is shown to be a mechanism for controlling structural complexity and, consequently, hypothesis-class growth.
- Numerically stable operators: The framework indicates that the use of numerically stable operators contributes to managing the sensitivity of predictions to parameter perturbations.
- Interval arithmetic: Similar to stable operators, interval arithmetic is linked to mechanisms that control the sensitivity of predictions to parameter perturbations.
- The analysis explicitly links these practical design considerations in GP to complexity terms within the proposed generalization bound, providing a principled explanation for observed empirical behaviors in GP-based SR.
Why This Matters
This work contributes to a more rigorous theoretical understanding of generalization properties in symbolic regression utilizing genetic programming. By providing a principled explanation for empirical practices through a decomposed generalization bound, it enhances the theoretical foundation for GP-based SR and its observed effectiveness in discovering interpretable mathematical expressions.