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Leveraging Black-box Models to Assess Feature Importance in Unconditional Distribution

Jing Zhou, Chunlin Li

TL;DR

The paper tackles the challenge of assessing feature influence on the unconditional distribution of an outcome when using pretrained black-box predictors. It defines the feature-importance curve β(τ) through a distributional, Von Mises-type expansion and provides a post hoc plug-in estimator that leverages a pretrained predictor without retraining, augmented by density extrapolation for tail estimation. A sparsification mechanism via stepwise backward pruning across a grid of quantiles yields a sparse, interpretable set of features contributing to different parts of the distribution. Empirical results on synthetic and high-dimensional data demonstrate faithful, sparse β(τ) estimates and computational efficiency, with limitations primarily under heavy-tailed error distributions.

Abstract

Understanding how changes in explanatory features affect the unconditional distribution of the outcome is important in many applications. However, existing black-box predictive models are not readily suited for analyzing such questions. In this work, we develop an approximation method to compute the feature importance curves relevant to the unconditional distribution of outcomes, while leveraging the power of pre-trained black-box predictive models. The feature importance curves measure the changes across quantiles of outcome distribution given an external impact of change in the explanatory features. Through extensive numerical experiments and real data examples, we demonstrate that our approximation method produces sparse and faithful results, and is computationally efficient.

Leveraging Black-box Models to Assess Feature Importance in Unconditional Distribution

TL;DR

The paper tackles the challenge of assessing feature influence on the unconditional distribution of an outcome when using pretrained black-box predictors. It defines the feature-importance curve β(τ) through a distributional, Von Mises-type expansion and provides a post hoc plug-in estimator that leverages a pretrained predictor without retraining, augmented by density extrapolation for tail estimation. A sparsification mechanism via stepwise backward pruning across a grid of quantiles yields a sparse, interpretable set of features contributing to different parts of the distribution. Empirical results on synthetic and high-dimensional data demonstrate faithful, sparse β(τ) estimates and computational efficiency, with limitations primarily under heavy-tailed error distributions.

Abstract

Understanding how changes in explanatory features affect the unconditional distribution of the outcome is important in many applications. However, existing black-box predictive models are not readily suited for analyzing such questions. In this work, we develop an approximation method to compute the feature importance curves relevant to the unconditional distribution of outcomes, while leveraging the power of pre-trained black-box predictive models. The feature importance curves measure the changes across quantiles of outcome distribution given an external impact of change in the explanatory features. Through extensive numerical experiments and real data examples, we demonstrate that our approximation method produces sparse and faithful results, and is computationally efficient.

Paper Structure

This paper contains 18 sections, 1 theorem, 27 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Quantile Regression.
  5. Sparse Learning and Model agnostic Feature Importance for explaining Black-Box models.
  6. Transfer Learning and Causal Invariance Principle.
  7. Problem Formulation
  8. Method
  9. Method for Black Box with Univariate Output
  10. Estimating $\beta(\tau)$ and Challenges.
  11. Error Density Estimation via Extrapolation
  12. Sparsified Feature Importance via Stepwise Backward Pruning
  13. Numerical Performance
  14. Synthetic Experiments
  15. Low-dimensional Settings
  16. ...and 3 more sections

Key Result

Lemma 1

Let $G = (1 -t)F + t F'$ where $F$ and $F'$ are two distribution functions and $t \in [0, 1]$. The functional $\nu(G)$ around $F$ can be approximated as where the remainder term $o_\nu$ is of order $t$ and depends on the functional $\nu$, and $\hbox{IF}(x; \nu)$ refers to the influence function at $F$.

Figures (2)

  • Figure 1: Illustration of $\beta(\tau)$. Red points are the original distribution of $(X, Y)$. Blue points are the counterfactual distribution after a shift intervention on $X$. Grey line is $\mathbb{E}(Y\mid X)$. $\beta(0.2) > 0$ but $\beta(0.8)\approx 0$.
  • Figure 2: The percentage of out-of-range observations against quantile level $\tau$. The data are generated from $Y = 1 -2 X_1 + 5X_2 + \varepsilon$, where $\varepsilon \sim N(0,1)$, $X = (X_1, \ldots, X_4)^\top \sim N(\bm 0_4, \Sigma)$, and $\Sigma_{ij} = 0.5^{i - j}$.

Theorems & Definitions (2)

  • Definition 1: Influence function
  • Lemma 1: von Mises linear approximation