Importance Sampling for Nonlinear Models

TL;DR

This paper extends importance sampling techniques to nonlinear models using the adjoint operator, enabling efficient training, explainability, and outlier detection with theoretical guarantees and experimental validation.

Contribution

It introduces a novel framework for importance sampling in nonlinear models via the adjoint operator, generalizing norm and leverage scores.

Findings

Sampling based on nonlinear scores offers approximation guarantees.

The method reduces computational complexity in training nonlinear models.

Experimental results validate the effectiveness across supervised learning tasks.

Abstract

While norm-based and leverage-score-based methods have been extensively studied for identifying "important" data points in linear models, analogous tools for nonlinear models remain significantly underdeveloped. By introducing the concept of the adjoint operator of a nonlinear map, we address this gap and generalize norm-based and leverage-score-based importance sampling to nonlinear settings. We demonstrate that sampling based on these generalized notions of norm and leverage scores provides approximation guarantees for the underlying nonlinear mapping, similar to linear subspace embeddings. As direct applications, these nonlinear scores not only reduce the computational complexity of training nonlinear models by enabling efficient sampling over large datasets but also offer a novel mechanism for model explainability and outlier detection. Our contributions are supported by both theoretical analyses and experimental results across a variety of supervised learning scenarios.