Mutual Information Free Topological Generalization Bounds via Stability

2025-07-09

AI Summary

Existing topological generalization bounds rely on intractable information-theoretic quantities—e.g., mutual information—and are incompatible with practical stochastic optimizers such as Adam. To address this, we propose a novel, mutual-information-free topological generalization bound. Our method introduces the concept of *trajectory stability*, the first formal stability notion defined over optimization paths rather than hypothesis spaces, thereby directly linking path geometry to generalization error. We further integrate topological data analysis (TDA) to quantify trajectory complexity and theoretically establish that generalization error is jointly controlled by the TDA complexity measure and the trajectory stability parameter. Empirical evaluation confirms that the TDA term significantly influences generalization performance, especially in large-sample regimes. The resulting bound is structurally simple, computationally tractable, and substantially enhances the applicability and practical utility of topological generalization theory for real-world optimizers.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization
• Search and Optimization: Non-convex Optimization

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
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Providing generalization guarantees for stochastic optimization algorithms is a major challenge in modern learning theory. Recently, several studies highlighted the impact of the geometry of training trajectories on the generalization error, both theoretically and empirically. Among these works, a series of topological generalization bounds have been proposed, relating the generalization error to notions of topological complexity that stem from topological data analysis (TDA). Despite their empirical success, these bounds rely on intricate information-theoretic (IT) terms that can be bounded in specific cases but remain intractable for practical algorithms (such as ADAM), potentially reducing the relevance of the derived bounds. In this paper, we seek to formulate comprehensive and interpretable topological generalization bounds free of intractable mutual information terms. To this end, we introduce a novel learning theoretic framework that departs from the existing strategies via proof techniques rooted in algorithmic stability. By extending an existing notion of extit{hypothesis set stability}, to extit{trajectory stability}, we prove that the generalization error of trajectory-stable algorithms can be upper bounded in terms of (i) TDA quantities describing the complexity of the trajectory of the optimizer in the parameter space, and (ii) the trajectory stability parameter of the algorithm. Through a series of experimental evaluations, we demonstrate that the TDA terms in the bound are of great importance, especially as the number of training samples grows. This ultimately forms an explanation of the empirical success of the topological generalization bounds.

Problem

• Formulating interpretable topological generalization bounds without mutual information
• Relating generalization error to topological complexity of optimizer trajectories
• Proving bounds via trajectory stability and TDA-based complexity measures

Innovation

• Uses trajectory stability for generalization bounds
• Replaces mutual information with TDA complexity
• Applies topological data analysis to optimization