Robust Learning, Inference and Uncertainty Quantification via Contrast

Abstract

In this dissertation we develop tools for using contrast to improve trustworthiness in machine learning. We do this in two specific ways. We provide a new framework for producing counterfactuals which uses contrast to verify that they will have the intended effect without acting adversarially. We also propose a new fine-grained measure of aleatoric uncertainty and present a contrast based method for estimating this measure. First we address \textit{Counterfactuals}, which are modified inputs that lead to a different outcome and are an important tool for understanding the logic used by machine learning classifiers and how to change an undesirable classification. Even if a counterfactual changes a classifier's decision, however, it may not affect the true underlying class probabilities, i.e. the counterfactual may act like an adversarial attack and ``fool'' the classifier. We propose a new framework for creating modified inputs that change the true underlying probabilities in a beneficial way which we call \textit{Trustworthy Actionable Perturbations} (TAP). This includes a novel verification procedure using contrast to ensure that TAP change the true class probabilities instead of acting adversarially. Our framework also includes new cost, reward, and goal definitions that are better suited to effectuating change in the real world. We present PAC-learnability results for our verification procedure and theoretically analyze our new method for measuring reward. We also develop a methodology for creating TAP and compare our results to those achieved by previous counterfactual methods. Second, we propose a new and intuitive metric for aleatoric uncertainty quantification (UQ), the prevalence of class collisions defined as the same input being observed in different classes. We use the rate of class collisions to define the collision matrix, a novel and uniquely fine-grained measure of uncertainty. For a classification problem involving $K$ classes, the $K\times K$ collision matrix $S$ measures the inherent difficulty in distinguishing between each pair of classes. We discuss several applications of the collision matrix, establish its fundamental mathematical properties, and show its relationship with existing UQ methods, including the Bayes error rate (BER). We also address the new problem of estimating the collision matrix using one-hot labeled data by proposing a series of innovative techniques to estimate $S$. First, we learn a pair-wise contrastive model which accepts two inputs and determines if they belong to the same class. We then show that this contrastive model (which is PAC learnable) can be used to estimate the row Gramian matrix of $S$, defined as $G=SS^T$. Finally, we show that under reasonable assumptions, $G$ can be used to uniquely recover $S$, a new result on non-negative matrices which could be of independent interest. With a method to estimate $S$ established, we demonstrate how this estimate of $S$, in conjunction with the contrastive model, can be used to estimate the posterior class portability distribution of any input. Experimental results are also presented to validate our methods of estimating the collision matrix and class posterior distributions on several datasets.