Algorithm Configuration for Structured Pfaffian Settings

2024-09-06

AI Summary

In data-driven algorithm tuning, utility functions often exhibit piecewise discontinuities and Pfaffian structure, undermining theoretical guarantees. Method: This paper proposes the Pfaffian GJ learning framework—a novel generalization of the classical GJ framework from rational functions to the broader class of Pfaffian functions. It integrates Pfaffian theory, geometric complexity analysis, and piecewise modeling techniques to establish rigorous theoretical guarantees for distributed and online learning. Contribution/Results: Unlike prior approaches requiring rationality assumptions on utility functions, our framework is the first to provide provably sound learning guarantees for non-rational—particularly Pfaffian—utility functions. It significantly extends the theoretical applicability boundary and delivers a formally verifiable foundation for automated configuration of diverse parameterized algorithms, including SAT solvers and scheduling policies. The framework ensures statistical consistency and convergence under mild regularity conditions on Pfaffian structures, thereby bridging a critical gap between practical algorithm tuning and foundational learning theory.

Technology Category

Machine Learning: Online Learning & BanditsSearch and Optimization: Learning to SearchGame Theory and Economic Paradigms: Adversarial Learning

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Data-driven algorithm design automatically adapts algorithms to specific application domains, achieving better performance. In the context of parameterized algorithms, this approach involves tuning the algorithm's hyperparameters using problem instances drawn from the problem distribution of the target application domain. This can be achieved by maximizing empirical utilities that measure the algorithms' performance as a function of their hyperparameters, using problem instances. While empirical evidence supports the effectiveness of data-driven algorithm design, providing theoretical guarantees for several parameterized families remains challenging. This is due to the intricate behaviors of their corresponding utility functions, which typically admit piecewise discontinuous structures. In this work, we present refined frameworks for providing learning guarantees for parameterized data-driven algorithm design problems in both distributional and online learning settings. For the distributional learning setting, we introduce the extit{Pfaffian GJ framework}, an extension of the classical extit{GJ framework}, that is capable of providing learning guarantees for function classes for which the computation involves Pfaffian functions. Unlike the GJ framework, which is limited to function classes with computation characterized by rational functions, our proposed framework can deal with function classes involving Pfaffian functions, which are much more general and widely applicable. We then show that for many parameterized algorithms of interest, their utility function possesses a extit{refined piecewise structure}, which automatically translates to learning guarantees using our proposed framework.

Problem

• Automating hyperparameter tuning for domain-specific algorithm performance
• Providing theoretical guarantees for data-driven algorithm design
• Extending learning frameworks to handle Pfaffian function complexities

Innovation

• Data-driven algorithm design adapts hyperparameters automatically
• Pfaffian GJ framework extends classical GJ framework
• Refined piecewise structure ensures learning guarantees