Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning

2026-02-03

AI Summary

This work addresses the limitations of traditional homotopy methods, which rely on handcrafted heuristics for step-size selection and termination criteria, resulting in poor generalization and suboptimal performance. The authors propose the Neural Predictor-Corrector (NPC), a novel framework that, for the first time, unifies diverse homotopy problems into a common sequential decision-making formulation. By integrating reinforcement learning with amortized training, NPC automatically learns a general-purpose solving strategy that transfers across tasks. Empirical evaluations demonstrate that NPC consistently outperforms both classical and specialized baselines across four distinct homotopy problem classes, exhibiting superior generalization, computational efficiency, and numerical stability.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The Homotopy paradigm, a general principle for solving challenging problems, appears across diverse domains such as robust optimization, global optimization, polynomial root-finding, and sampling. Practical solvers for these problems typically follow a predictor-corrector (PC) structure, but rely on hand-crafted heuristics for step sizes and iteration termination, which are often suboptimal and task-specific. To address this, we unify these problems under a single framework, which enables the design of a general neural solver. Building on this unified view, we propose Neural Predictor-Corrector (NPC), which replaces hand-crafted heuristics with automatically learned policies. NPC formulates policy selection as a sequential decision-making problem and leverages reinforcement learning to automatically discover efficient strategies. To further enhance generalization, we introduce an amortized training mechanism, enabling one-time offline training for a class of problems and efficient online inference on new instances. Experiments on four representative homotopy problems demonstrate that our method generalizes effectively to unseen instances. It consistently outperforms classical and specialized baselines in efficiency while demonstrating superior stability across tasks, highlighting the value of unifying homotopy methods into a single neural framework.

Problem

• Homotopy
• Predictor-Corrector
• Heuristics
• Step Size
• Iteration Termination

Innovation

• Neural Predictor-Corrector
• Homotopy methods
• Reinforcement learning
• Amortized training
• Unified framework