Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning

TL;DR

This paper introduces Neural Predictor-Corrector (NPC), a reinforcement learning-based neural framework that automates and improves the efficiency and stability of homotopy problem solvers across various domains.

Contribution

It unifies homotopy problem solving under a neural, reinforcement learning-based framework, replacing heuristics with learned policies for better generalization and performance.

Findings

NPC outperforms classical solvers in efficiency.

NPC demonstrates superior stability across tasks.

Effective generalization to unseen problem instances.

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.