Reliable and Scalable Variable Importance Estimation via Warm-start and Early Stopping

As opaque black-box predictive models such as neural networks become more prevalent, the need to develop interpretations for these models is of great interest. The concept of $\textit{variable importance}$ is an interpretability measure that applies to any predictive model and assesses how much a variable or set of variables improves prediction performance. When the number of variables is large, estimating variable importance presents a significant challenge because re-training neural networks or other black-box algorithms requires significant additional computation. In this paper, we address this challenge for algorithms using gradient descent and gradient boosting (e.g. neural networks, gradient-boosted decision trees). By using the ideas of early stopping of gradient-based methods in combination with warm-start using the $\textit{dropout}$ method, we develop a scalable method to estimate variable importance for any algorithm that can be expressed as an $\textit{iterative kernel update equation}$. Importantly, we provide theoretical guarantees by using the theory for early stopping of kernel-based methods for neural networks with sufficient large width and gradient-boosting decision trees that use symmetric tree as a weaker learner. We also demonstrate the efficacy of our methods through simulations and a real data example which illustrates the computational benefit of early stopping rather than fully re-training the model as well as the increased accuracy of taking initial steps from the dropout solution.