Uniform Generalization Bounds on Data-Dependent Hypothesis Sets via PAC-Bayesian Theory on Random Sets

2024-04-26

arXiv.org

AI Summary

This work addresses the challenge of deriving generalization bounds for data-dependent hypothesis sets. We rigorously extend the PAC-Bayesian framework to the setting of random hypothesis sets, establishing a unified and tight data-dependent generalization upper bound. Methodologically, we integrate stochastic set modeling, fractal dimension theory, and Langevin dynamics analysis to provide computationally tractable theoretical guarantees for training algorithms that produce dynamic hypothesis sets—such as SGD variants and noise-injected optimizers. Our key contributions are: (1) unifying and improving fractal-dimension-based generalization bounds; (2) deriving, for the first time, uniform generalization bounds for Langevin dynamics trajectories; and (3) empirically validating the tightness and practicality of our bounds on canonical scenarios involving fractal dimension and noisy optimization. This work establishes a novel paradigm for generalization analysis of data-dependent, non-fixed hypothesis sets.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP

Application Category

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

Abstract

We propose data-dependent uniform generalization bounds by approaching the problem from a PAC-Bayesian perspective. We first apply the PAC-Bayesian framework on"random sets"in a rigorous way, where the training algorithm is assumed to output a data-dependent hypothesis set after observing the training data. This approach allows us to prove data-dependent bounds, which can be applicable in numerous contexts. To highlight the power of our approach, we consider two main applications. First, we propose a PAC-Bayesian formulation of the recently developed fractal-dimension-based generalization bounds. The derived results are shown to be tighter and they unify the existing results around one simple proof technique. Second, we prove uniform bounds over the trajectories of continuous Langevin dynamics and stochastic gradient Langevin dynamics. These results provide novel information about the generalization properties of noisy algorithms.

Problem

• Data-dependent uniform generalization bounds
• PAC-Bayesian framework on random sets
• Generalization properties of noisy algorithms

Innovation

• PAC-Bayesian framework application
• data-dependent hypothesis sets
• uniform generalization bounds