Proofs as Explanations: Short Certificates for Reliable Predictions

• URL: http://arxiv.org/abs/2504.08377v3
• Date: Tue, 24 Jun 2025 19:55:51 GMT
• Title: Proofs as Explanations: Short Certificates for Reliable Predictions
• Authors: Avrim Blum, Steve Hanneke, Chirag Pabbaraju, Donya Saless,
• Abstract summary: We consider a model for explainable AI in which an explanation for a prediction $h(x)=y$ consists of a subset $S'$ of the training data.<n>A set $S'$ of size $d+1$ could be released as an explanation for a positive prediction, and would serve as a short proof of correctness of the prediction under the assumption of realizability.
• Score: 25.157401709060064
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider a model for explainable AI in which an explanation for a prediction $h(x)=y$ consists of a subset $S'$ of the training data (if it exists) such that all classifiers $h' \in H$ that make at most $b$ mistakes on $S'$ predict $h'(x)=y$. Such a set $S'$ serves as a proof that $x$ indeed has label $y$ under the assumption that (1) the target function $h^\star$ belongs to $H$, and (2) the set $S$ contains at most $b$ corrupted points. For example, if $b=0$ and $H$ is the family of linear classifiers in $\mathbb{R}^d$, and if $x$ lies inside the convex hull of the positive data points in $S$ (and hence every consistent linear classifier labels $x$ as positive), then Carath\'eodory's theorem states that $x$ lies inside the convex hull of $d+1$ of those points. So, a set $S'$ of size $d+1$ could be released as an explanation for a positive prediction, and would serve as a short proof of correctness of the prediction under the assumption of realizability. In this work, we consider this problem more generally, for general hypothesis classes $H$ and general values $b\geq 0$. We define the notion of the robust hollow star number of $H$ (which generalizes the standard hollow star number), and show that it precisely characterizes the worst-case size of the smallest certificate achievable, and analyze its size for natural classes. We also consider worst-case distributional bounds on certificate size, as well as distribution-dependent bounds that we show tightly control the sample size needed to get a certificate for any given test example. In particular, we define a notion of the certificate coefficient $\varepsilon_x$ of an example $x$ with respect to a data distribution $D$ and target function $h^\star$, and prove matching upper and lower bounds on sample size as a function of $\varepsilon_x$, $b$, and the VC dimension $d$ of $H$.

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