Fine-Grained Uncertainty Quantification via Collisions

We propose a new approach for fine-grained uncertainty quantification (UQ) using a collision matrix. For a classification problem involving $K$ classes, the $K\times K$ collision matrix $S$ measures the inherent (aleatoric) difficulty in distinguishing between each pair of classes. In contrast to existing UQ methods, the collision matrix gives a much more detailed picture of the difficulty of classification. We discuss several possible downstream applications of the collision matrix, establish its fundamental mathematical properties, as well as show its relationship with existing UQ methods, including the Bayes error rate. We also address the new problem of estimating the collision matrix using one-hot labeled data. We propose a series of innovative techniques to estimate $S$. First, we learn a contrastive binary classifier which takes two inputs and determines if they belong to the same class. We then show that this contrastive classifier (which is PAC learnable) can be used to reliably estimate the Gramian matrix of $S$, defined as $G=S^TS$. Finally, we show that under very mild assumptions, $G$ can be used to uniquely recover $S$, a new result on stochastic matrices which could be of independent interest. Experimental results are also presented to validate our methods on several datasets.

Trustworthy Actionable Perturbations

Counterfactuals, or modified inputs that lead to a different outcome, 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 Trustworthy Actionable Perturbations (TAP). This includes a novel verification procedure 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.