Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning

TL;DR

This work unifies diverse homotopy-based problem solve methods under a single predictor-corrector framework and replaces hand-crafted heuristics with learned policies via reinforcement learning. The Neural Predictor-Corrector (NPC) treats predictor step size and corrector tolerance as actions in a Markov decision process and trains an amortized policy with PPO to generalize across problem instances. Across four representative tasks—robust optimization, global optimization, polynomial root-finding, and sampling—NPC achieves superior efficiency and stability compared to classical baselines and generalizes to unseen instances without per-instance training. The approach promises practical impact by enabling training-free deployment of a general, efficient solver for a broad class of challenging problems that can be framed as homotopy continuations.

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.

Figures (5)

• Figure 1: Homotopy paradigm across domains. The homotopy interpolation (blue loss functions in optimization, green polynomial roots in polynomial root-finding, and red probability densities in sampling) is explicitly defined, while the inner solution trajectory (orange curve) must be implicitly tracked.
• Figure 2: Illustration of the Predictor-Corrector algorithm.Predictor proposes the next level and provides an initial solution estimate, while Corrector iteratively refines this estimate to project it back onto the solution trajectory. Orange curve denotes the implicit solution trajectory, as in Fig. \ref{['fig:homotopy_problem']}.
• Figure 3: RL formulation of the proposed Neural Predictor-Corrector (NPC). At each homotopy level, the agent observes the current state (including homotopy level, corrector statistics, and convergence velocity), outputs actions that adapt the predictor’s step size and the corrector’s tolerance, and receives rewards designed to balance accuracy and efficiency.
• Figure 4: Trade-off between efficiency and precision. Efficiency is measured in terms of corrector iterations, and precision reflects solution accuracy, for NPC-accelerated versus classical methods.
• Figure 5: Supplementary box plots of performance metrics. These visualizations illustrate the result distributions over 50 independent trials, providing a more intuitive understanding of the stability and efficiency of each method.