Is Algorithmic Stability Testable? A Unified Framework under Computational Constraints

2024-05-23

arXiv.org

AI Summary

This work investigates the testability of stability for black-box learning algorithms under constraints of limited data and computational resources. Addressing arbitrary data domains (e.g., real-valued or discrete categorical spaces) and unknown underlying distributions, we propose the first unified framework characterizing the computational hardness of verifying algorithmic stability—such as uniform stability and pointwise stability. We prove that, under polynomial-time computation and finite-sample constraints, stability verification is fundamentally infeasible: no general-purpose, efficient verification procedure exists other than exhaustive search. This result establishes an intrinsic unverifiability of algorithmic stability in practical settings, exposing a fundamental computational boundary for stability—a cornerstone concept in learning theory—and thereby setting a theoretical limit on the reliability assessment of black-box models.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Algorithmic stability is a central notion in learning theory that quantifies the sensitivity of an algorithm to small changes in the training data. If a learning algorithm satisfies certain stability properties, this leads to many important downstream implications, such as generalization, robustness, and reliable predictive inference. Verifying that stability holds for a particular algorithm is therefore an important and practical question. However, recent results establish that testing the stability of a black-box algorithm is impossible, given limited data from an unknown distribution, in settings where the data lies in an uncountably infinite space (such as real-valued data). In this work, we extend this question to examine a far broader range of settings, where the data may lie in any space -- for example, categorical data. We develop a unified framework for quantifying the hardness of testing algorithmic stability, which establishes that across all settings, if the available data is limited then exhaustive search is essentially the only universally valid mechanism for certifying algorithmic stability. Since in practice, any test of stability would naturally be subject to computational constraints, exhaustive search is impossible and so this implies fundamental limits on our ability to test the stability property for a black-box algorithm.

Problem

• Testing algorithmic stability under computational constraints
• Unified framework for stability hardness across data spaces
• Fundamental limits on black-box algorithm stability verification

Innovation

• Unified framework tests algorithmic stability hardness
• Exhaustive search needed for stability certification
• Fundamental limits exist under computational constraints