# Symbolic Metamodels for Interpreting Black-boxes Using Primitive Functions

arXiv: https://arxiv.org/abs/2302.04791 · HTML: https://mlsanity.com/paper/2302.04791/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 46 references that are in the map.
- First posted on arXiv 9 Feb 2023.

## Builds on (most cited first, 21 of 21)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [This Looks Like That: Deep Learning for Interpretable Image Recognition](https://mlsanity.com/paper/1806.10574/) (2018)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Towards Robust Interpretability with Self-Explaining Neural Networks](https://mlsanity.com/paper/1806.07538/) (2018)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Learning to Explain: An Information-Theoretic Perspective on Model Interpretation](https://mlsanity.com/paper/1802.07814/) (2018)
- [Model-Agnostic Interpretability of Machine Learning](https://mlsanity.com/paper/1606.05386/) (2016)
- [Local Rule-Based Explanations of Black Box Decision Systems](https://mlsanity.com/paper/1805.10820/) (2018)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [Where are we now? A large benchmark study of recent symbolic regression methods](https://mlsanity.com/paper/1804.09331/) (2018)
- [Symbolic Pregression: Discovering Physical Laws from Distorted Video](https://mlsanity.com/paper/2005.11212/) (2020)
- [Interpreting Neural Network Judgments via Minimal, Stable, and Symbolic Corrections](https://mlsanity.com/paper/1802.07384/) (2018)
- [Learning outside the Black-Box: The pursuit of interpretable models](https://mlsanity.com/paper/2011.08596/) (2020)
- [GP-RVM: Genetic Programing-based Symbolic Regression Using Relevance Vector Machine](https://mlsanity.com/paper/1806.02502/) (2018)

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

- [Loss Balancing for Fair Supervised Learning](https://mlsanity.com/paper/2311.03714/) (2023)
- [Closed-Form Interpretation of Neural Network Classifiers with Symbolic Gradients](https://mlsanity.com/paper/2401.04978/) (2024)

## Abstract (arXiv)

One approach for interpreting black-box machine learning models is to find a global approximation of the model using simple interpretable functions, which is called a metamodel (a model of the model). Approximating the black-box with a metamodel can be used to 1) estimate instance-wise feature importance; 2) understand the functional form of the model; 3) analyze feature interactions. In this work, we propose a new method for finding interpretable metamodels. Our approach utilizes Kolmogorov superposition theorem, which expresses multivariate functions as a composition of univariate functions (our primitive parameterized functions). This composition can be represented in the form of a tree. Inspired by symbolic regression, we use a modified form of genetic programming to search over different tree configurations. Gradient descent (GD) is used to optimize the parameters of a given configuration. Our method is a novel memetic algorithm that uses GD not only for training numerical constants but also for the training of building blocks. Using several experiments, we show that our method outperforms recent metamodeling approaches suggested for interpreting black-boxes.