Proofs as Explanations: Short Certificates for Reliable Predictions

2025-04-11

AI Summary

This paper addresses the problem of constructing concise and robust certificates for prediction interpretability in AI: given a hypothesis class ℋ and a training set containing at most b noisy points, how to identify the smallest subset S′ such that all classifiers in ℋ with error ≤ b on S′ agree on the prediction for a target input x? To this end, we introduce the *robust hollow star number*—a novel combinatorial measure characterizing the minimal certificate size. We define a distribution-dependent certificate coefficient εₓ and derive tight upper and lower bounds on the sample complexity. By unifying VC-dimension theory, Carathéodory’s theorem, and combinatorial geometry, we jointly analyze hypothesis class structure and data distribution. Our framework precisely characterizes worst-case and average-case certificate lengths for general ℋ, yields explicit upper bounds for natural classes (e.g., linear separators), and provides efficiently checkable conditions for certificate existence.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $bgeq 0$. 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 $H$.

Problem

• Finding minimal training subsets certifying reliable AI predictions
• Generalizing certificate size analysis for arbitrary hypothesis classes
• Establishing sample complexity bounds for robust explanation certificates

Innovation

• Short certificates for reliable AI predictions
• Robust hollow star number generalizes hypothesis classes
• Matching bounds on sample size via certificate coefficient