# Lazy Estimation of Variable Importance for Large Neural Networks

arXiv: https://arxiv.org/abs/2207.09097 · HTML: https://mlsanity.com/paper/2207.09097/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 19 Jul 2022.

## Builds on (most cited first, 19 of 19)

- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [A Survey Of Methods For Explaining Black Box Models](https://mlsanity.com/paper/1802.01933/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [A Benchmark for Interpretability Methods in Deep Neural Networks](https://mlsanity.com/paper/1806.10758/) (2018)
- [Early stopping and non-parametric regression: An optimal data-dependent stopping rule](https://mlsanity.com/paper/1306.3574/) (2013)
- [Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks](https://mlsanity.com/paper/2012.11654/) (2020)
- [The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training](https://mlsanity.com/paper/2007.12826/) (2020)
- [Efficient nonparametric statistical inference on population feature importance using Shapley values](https://mlsanity.com/paper/2006.09481/) (2020)
- [A general framework for inference on algorithm-agnostic variable importance](https://mlsanity.com/paper/2004.03683/) (2020)
- [Investigating the fidelity of explainable artificial intelligence methods for applications of convolutional neural networks in geoscience](https://mlsanity.com/paper/2202.03407/) (2022)
- [A knockoff filter for high-dimensional selective inference](https://mlsanity.com/paper/1602.03574/) (2016)
- [Neural Network Attribution Methods for Problems in Geoscience: A Novel Synthetic Benchmark Dataset](https://mlsanity.com/paper/2103.10005/) (2021)
- [Floodgate: inference for model-free variable importance](https://mlsanity.com/paper/2007.01283/) (2020)
- [Dropout Feature Ranking for Deep Learning Models](https://mlsanity.com/paper/1712.08645/) (2017)

## Built on by (most cited first, 5 of 5)

- [Statistically Valid Variable Importance Assessment through Conditional Permutations](https://mlsanity.com/paper/2309.07593/) (2023)
- [Variable Importance in High-Dimensional Settings Requires Grouping](https://mlsanity.com/paper/2312.10858/) (2023)
- [MinShap: A Shapley-Based Framework for Feature Redundancy](https://mlsanity.com/paper/2604.15107/) (2026)
- [Fast, Distribution-free Predictive Inference for Neural Networks with Coverage Guarantees](https://mlsanity.com/paper/2306.06582/) (2023)
- [Comparing Model-agnostic Feature Selection Methods through Relative Efficiency](https://mlsanity.com/paper/2508.14268/) (2025)

## Abstract (arXiv)

As opaque predictive models increasingly impact many areas of modern life, interest in quantifying the importance of a given input variable for making a specific prediction has grown. Recently, there has been a proliferation of model-agnostic methods to measure variable importance (VI) that analyze the difference in predictive power between a full model trained on all variables and a reduced model that excludes the variable(s) of interest. A bottleneck common to these methods is the estimation of the reduced model for each variable (or subset of variables), which is an expensive process that often does not come with theoretical guarantees. In this work, we propose a fast and flexible method for approximating the reduced model with important inferential guarantees. We replace the need for fully retraining a wide neural network by a linearization initialized at the full model parameters. By adding a ridge-like penalty to make the problem convex, we prove that when the ridge penalty parameter is sufficiently large, our method estimates the variable importance measure with an error rate of O(frac1√n) where n is the number of training samples. We also show that our estimator is asymptotically normal, enabling us to provide confidence bounds for the VI estimates. We demonstrate through simulations that our method is fast and accurate under several data-generating regimes, and we demonstrate its real-world applicability on a seasonal climate forecasting example.