Genetic Programming Struggles to Efficiently Explore the Symbolic Regression Search Space

Core Concepts

Genetic programming explores only a small fraction of the unique expressions in the symbolic regression search space and repeatedly evaluates expressions that are semantically equivalent to previously visited ones.

Abstract

The paper analyzes the efficiency of genetic programming (GP) for symbolic regression (SR) tasks by comparing it to an exhaustive search method called Exhaustive Symbolic Regression (ESR). The key findings are:

ESR is able to fully enumerate the search space of unique expressions up to a certain length limit, revealing that GP only explores a small fraction of this space and repeatedly evaluates semantically equivalent expressions.

For two real-world datasets (flow in rough pipes and radial acceleration relation in galaxy dynamics), GP fails to find the globally optimal expressions found by ESR, even when allowing longer expressions.

The success probability of GP in finding good solutions is much lower than an idealized random search in the space of unique expressions, especially for stricter quality thresholds.

The analysis of expressions visited by GP shows that a large fraction (50-80%) are semantically equivalent to previously visited ones, indicating significant redundancy in the GP search.

The authors conclude that GP exhibits worrying inefficiency in exploring the SR search space, and that methods to better control redundancy and focus the search on unique expressions are needed to improve the performance of GP for symbolic regression.

Stats

The best MSE for the Nikuradse dataset is 0.0027 for expressions with up to 12 nodes.
The best negative log-likelihood for the RAR dataset is -1013.24 for expressions with up to 12 nodes.

Quotes

"The results show a worrying inefficiency of GP for SR with a very low rate of unique solutions and a success probability smaller than an idealised random search when limiting the search space to short expressions."
"The analysis is made possible by improved algorithms for equality saturation, which we use to improve the Exhaustive Symbolic Regression algorithm; this produces the set of semantically unique expression structures, orders of magnitude smaller than the full symbolic regression search space."

Key Insights Distilled From

The Inefficiency of Genetic Programming for Symbolic Regression -- Extended Version

by Gabriel Kron... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17292.pdf

Deeper Inquiries

How can genetic programming be modified to better explore the space of unique expressions and avoid revisiting semantically equivalent solutions?

Genetic programming can be modified in several ways to improve the exploration of unique expressions and prevent revisiting semantically equivalent solutions:

Diversity Maintenance: Implement mechanisms to maintain diversity in the population by introducing strategies like niche formation, speciation, or crowding to prevent premature convergence to suboptimal solutions.

Novelty Search: Incorporate novelty search into the genetic programming algorithm to encourage the discovery of novel and diverse solutions rather than focusing solely on fitness improvement. This can help in exploring a wider range of the search space.

Fitness Landscape Analysis: Utilize techniques to analyze the fitness landscape of the problem space to guide the search towards regions with promising solutions and avoid getting stuck in local optima.

Dynamic Parameter Adjustment: Implement adaptive mechanisms to adjust parameters such as mutation rates, crossover probabilities, and population sizes dynamically during the evolution process to balance exploration and exploitation effectively.

Semantic Simplification: Integrate semantic simplification techniques to identify and eliminate semantically equivalent expressions, reducing redundancy in the search space and improving the efficiency of the search process.

Advanced Crossover and Mutation Operators: Develop specialized crossover and mutation operators that promote the generation of diverse and unique solutions, potentially incorporating domain-specific knowledge to guide the search.

By incorporating these modifications, genetic programming can enhance its ability to explore the space of unique expressions more effectively and avoid revisiting semantically equivalent solutions, leading to improved search efficiency and better performance in symbolic regression tasks.

How can genetic programming be modified to better explore the space of unique expressions and avoid revisiting semantically equivalent solutions?

Genetic programming can be modified in several ways to improve the exploration of unique expressions and prevent revisiting semantically equivalent solutions:

Diversity Maintenance: Implement mechanisms to maintain diversity in the population by introducing strategies like niche formation, speciation, or crowding to prevent premature convergence to suboptimal solutions.

Novelty Search: Incorporate novelty search into the genetic programming algorithm to encourage the discovery of novel and diverse solutions rather than focusing solely on fitness improvement. This can help in exploring a wider range of the search space.

Fitness Landscape Analysis: Utilize techniques to analyze the fitness landscape of the problem space to guide the search towards regions with promising solutions and avoid getting stuck in local optima.

Dynamic Parameter Adjustment: Implement adaptive mechanisms to adjust parameters such as mutation rates, crossover probabilities, and population sizes dynamically during the evolution process to balance exploration and exploitation effectively.

Semantic Simplification: Integrate semantic simplification techniques to identify and eliminate semantically equivalent expressions, reducing redundancy in the search space and improving the efficiency of the search process.

Advanced Crossover and Mutation Operators: Develop specialized crossover and mutation operators that promote the generation of diverse and unique solutions, potentially incorporating domain-specific knowledge to guide the search.

By incorporating these modifications, genetic programming can enhance its ability to explore the space of unique expressions more effectively and avoid revisiting semantically equivalent solutions, leading to improved search efficiency and better performance in symbolic regression tasks.

How can genetic programming be modified to better explore the space of unique expressions and avoid revisiting semantically equivalent solutions?

Genetic programming can be modified in several ways to improve the exploration of unique expressions and prevent revisiting semantically equivalent solutions:

Diversity Maintenance: Implement mechanisms to maintain diversity in the population by introducing strategies like niche formation, speciation, or crowding to prevent premature convergence to suboptimal solutions.

Novelty Search: Incorporate novelty search into the genetic programming algorithm to encourage the discovery of novel and diverse solutions rather than focusing solely on fitness improvement. This can help in exploring a wider range of the search space.

Fitness Landscape Analysis: Utilize techniques to analyze the fitness landscape of the problem space to guide the search towards regions with promising solutions and avoid getting stuck in local optima.

Dynamic Parameter Adjustment: Implement adaptive mechanisms to adjust parameters such as mutation rates, crossover probabilities, and population sizes dynamically during the evolution process to balance exploration and exploitation effectively.

Semantic Simplification: Integrate semantic simplification techniques to identify and eliminate semantically equivalent expressions, reducing redundancy in the search space and improving the efficiency of the search process.

Advanced Crossover and Mutation Operators: Develop specialized crossover and mutation operators that promote the generation of diverse and unique solutions, potentially incorporating domain-specific knowledge to guide the search.

By incorporating these modifications, genetic programming can enhance its ability to explore the space of unique expressions more effectively and avoid revisiting semantically equivalent solutions, leading to improved search efficiency and better performance in symbolic regression tasks.