Quantile-constrained Wasserstein projections for robust interpretability of numerical and machine learning models

TL;DR

This paper introduces a unified framework combining uncertainty quantification and machine learning interpretability by using quantile-constrained Wasserstein projections to assess model robustness against input perturbations.

Contribution

It proposes a novel approach for robustness analysis using Wasserstein projections with quantile constraints, applicable to both UQ and ML models, and provides analytical solutions and smoothing techniques.

Findings

Analytical solutions for perturbation problems under Wasserstein distance.

Smoothing perturbations via isotonic polynomial approximations.

Numerical experiments demonstrate computational feasibility and robustness insights.

Abstract

Robustness studies of black-box models is recognized as a necessary task for numerical models based on structural equations and predictive models learned from data. These studies must assess the model's robustness to possible misspecification of regarding its inputs (e.g., covariate shift). The study of black-box models, through the prism of uncertainty quantification (UQ), is often based on sensitivity analysis involving a probabilistic structure imposed on the inputs, while ML models are solely constructed from observed data. Our work aim at unifying the UQ and ML interpretability approaches, by providing relevant and easy-to-use tools for both paradigms. To provide a generic and understandable framework for robustness studies, we define perturbations of input information relying on quantile constraints and projections with respect to the Wasserstein distance between probability measures, while preserving their dependence structure. We show that this perturbation problem can be analytically solved. Ensuring regularity constraints by means of isotonic polynomial approximations leads to smoother perturbations, which can be more suitable in practice. Numerical experiments on real case studies, from the UQ and ML fields, highlight the computational feasibility of such studies and provide local and global insights on the robustness of black-box models to input perturbations.