# Local Optimization Often is Ill-conditioned in Genetic Programming for Symbolic Regression

arXiv: https://arxiv.org/abs/2209.00942 · HTML: https://mlsanity.com/paper/2209.00942/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2022.

## Built on by (most cited first, 2 of 2)

- [Prediction Intervals and Confidence Regions for Symbolic Regression Models based on Likelihood Profiles](https://mlsanity.com/paper/2209.06454/) (2022)
- [Efficient Constant Optimization for Symbolic Regression with GPU-Accelerated Tree-Based Genetic Programming](https://mlsanity.com/paper/2609.03352/) (2026)

## Abstract (arXiv)

Gradient-based local optimization has been shown to improve results of genetic programming (GP) for symbolic regression. Several state-of-the-art GP implementations use iterative nonlinear least squares (NLS) algorithms such as the Levenberg-Marquardt algorithm for local optimization. The effectiveness of NLS algorithms depends on appropriate scaling and conditioning of the optimization problem. This has so far been ignored in symbolic regression and GP literature. In this study we use a singular value decomposition of NLS Jacobian matrices to determine the numeric rank and the condition number. We perform experiments with a GP implementation and six different benchmark datasets. Our results show that rank-deficient and ill-conditioned Jacobian matrices occur frequently and for all datasets. The issue is less extreme when restricting GP tree size and when using many non-linear functions in the function set.