Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explainable Learning with Gaussian Processes

Kurt Butler, Guanchao Feng, Petar M. Djuric

TL;DR

This work takes a principled approach to defining attributions under model uncertainty, extending the existing literature, and shows that although GPR is a highly flexible and non-parametric approach, it can derive interpretable, closed-form expressions for the feature attributions.

Abstract

The field of explainable artificial intelligence (XAI) attempts to develop methods that provide insight into how complicated machine learning methods make predictions. Many methods of explanation have focused on the concept of feature attribution, a decomposition of the model's prediction into individual contributions corresponding to each input feature. In this work, we explore the problem of feature attribution in the context of Gaussian process regression (GPR). We take a principled approach to defining attributions under model uncertainty, extending the existing literature. We show that although GPR is a highly flexible and non-parametric approach, we can derive interpretable, closed-form expressions for the feature attributions. When using integrated gradients as an attribution method, we show that the attributions of a GPR model also follow a Gaussian process distribution, which quantifies the uncertainty in attribution arising from uncertainty in the model. We demonstrate, both through theory and experimentation, the versatility and robustness of this approach. We also show that, when applicable, the exact expressions for GPR attributions are both more accurate and less computationally expensive than the approximations currently used in practice. The source code for this project is freely available under MIT license at https://github.com/KurtButler/2024_attributions_paper.

Explainable Learning with Gaussian Processes

TL;DR

This work takes a principled approach to defining attributions under model uncertainty, extending the existing literature, and shows that although GPR is a highly flexible and non-parametric approach, it can derive interpretable, closed-form expressions for the feature attributions.

Abstract

The field of explainable artificial intelligence (XAI) attempts to develop methods that provide insight into how complicated machine learning methods make predictions. Many methods of explanation have focused on the concept of feature attribution, a decomposition of the model's prediction into individual contributions corresponding to each input feature. In this work, we explore the problem of feature attribution in the context of Gaussian process regression (GPR). We take a principled approach to defining attributions under model uncertainty, extending the existing literature. We show that although GPR is a highly flexible and non-parametric approach, we can derive interpretable, closed-form expressions for the feature attributions. When using integrated gradients as an attribution method, we show that the attributions of a GPR model also follow a Gaussian process distribution, which quantifies the uncertainty in attribution arising from uncertainty in the model. We demonstrate, both through theory and experimentation, the versatility and robustness of this approach. We also show that, when applicable, the exact expressions for GPR attributions are both more accurate and less computationally expensive than the approximations currently used in practice. The source code for this project is freely available under MIT license at https://github.com/KurtButler/2024_attributions_paper.
Paper Structure (34 sections, 7 theorems, 105 equations, 7 figures)

This paper contains 34 sections, 7 theorems, 105 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. The Feature Attribution Problem
  5. Motivation: Bayesian Linear Regression
  6. Integrated Gradients
  7. Geometric Interpretation
  8. Axiomatic Interpretation
  9. Linear Operators
  10. Gaussian Process Regression
  11. Covariance Functions
  12. Derivatives and Integrals of GPs
  13. Spectral Approximations to GPR
  14. Attributions in GPR Models
  15. SE and ARD-SE Kernels
  16. ...and 19 more sections

Key Result

Lemma 1

If $F \sim \mathcal{GP}(m,k)$, where $m\in C^1(\mathbb{R}^D)$ and $k\in C^2 (\mathbb{R}^D \times \mathbb{R}^D)$, then

Figures (7)

  • Figure 1: A demonstration that the uncertainty in an attribution grows with the distance from the baseline, using data simulated with the model in \ref{['eq:syntheticdatamodel']}. The baseline is $\tilde{\mathbf{x}}=\mathbf{0}$ and we compute attributions for predictions along the path $\boldsymbol{\beta}(t)=t[1,1]^\top$.
  • Figure 2: Feature attribution for patients from the breast cancer data set. Predictions were made using a GPR model with ARD-SE kernel.
  • Figure 3: Feature attributions for a prediction using the Taipei Housing Data.
  • Figure 4: Convergence of attributions with the approximated integral, \ref{['eq:approxIG']}, to the exact attributions, as the number of partitions $L$ increases.
  • Figure 5: Attributions to alcohol in the Wine quality data set. As the number of frequencies used in the RFGP approximation, $M$, increases, the densities of the RFGP attributions more closely align with the exact GPR attributions. We show both individual realizations of the RFGP model (the blue and red curves), and a marginalized model which integrates over 500 independent RFGP models (the red shaded region).
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1: Derivatives of GPs
  • Lemma 2: Integral of a GP
  • Theorem 1: Integrated gradients preserve Gaussianity
  • Corollary 1: Attributions of the GPR Posterior-Predictive
  • Lemma 3: Scaling preserves Gaussianity
  • Lemma 4: Precompositions preserve Gaussianity
  • Proposition 1