Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning

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.

💡 Research Summary

The paper tackles a fundamental limitation of existing homotopy solvers, which typically follow a predictor‑corrector (PC) scheme but rely on hand‑crafted heuristics for step‑size selection and termination criteria. These heuristics are problem‑specific, often sub‑optimal, and hinder generalization across instances. The authors first unify four seemingly disparate domains—robust optimization (Graduated Non‑Convexity), global optimization (Gaussian homotopy), polynomial root‑finding (homotopy continuation), and high‑dimensional sampling (Annealed Langevin Dynamics)—under a single mathematical framework. In each case a homotopy interpolation H(x, t) connects a simple source problem H(x, 0) with a complex target H(x, 1); the solution trajectory x*(t) must be traced as t progresses from 0 to 1. The PC algorithm decomposes this tracing into (i) a predictor that proposes the next homotopy level and an initial estimate, and (ii) a corrector that iteratively refines the estimate until a convergence criterion is met.

To replace static heuristics, the authors formulate the entire PC process as a Markov Decision Process (MDP). The state s at level n encodes the current homotopy level tₙ₋₁, the previous corrector tolerance εₙ₋₁, the number of corrector iterations iₙ₋₁, and a convergence‑velocity metric τₙ₋₁ (e.g., relative change in objective value or statistical distance). The action a consists of two components: a step‑size Δtₙ that determines how far the predictor moves forward, and a corrector termination policy (either a tolerance εₙ or a maximum iteration count iₙ^max). The reward combines accuracy (final objective error, root‑finding residual, or sampling discrepancy) with efficiency (computation time, number of corrector steps), encouraging policies that achieve high precision with minimal work.

Learning is performed via reinforcement learning (policy‑gradient methods) on a distribution of problem instances. Crucially, the authors adopt an amortized training regime: a single offline training phase produces a policy that can be deployed on any new instance from the same problem class without further fine‑tuning. This contrasts with prior learning‑based homotopy methods that required per‑instance training.

Experiments on the four representative tasks demonstrate the efficacy of the Neural Predictor‑Corrector (NPC) framework. In robust optimization (GNC) NPC automatically adjusts the non‑convexity schedule, outperforming fixed schedules by reducing the number of Levenberg‑Marquardt iterations while maintaining or improving registration accuracy. In Gaussian homotopy for global optimization, NPC learns to shrink the kernel bandwidth more aggressively when the landscape is smooth and to take smaller steps near sharp transitions, achieving 30‑40 % fewer gradient evaluations than baseline annealing. For polynomial continuation, NPC’s adaptive step‑size prevents path‑following failures that plague static step schemes, yielding faster convergence to all target roots. In annealed Langevin dynamics, NPC balances the number of Langevin steps and temperature schedule, leading to higher sample quality (lower Kernelized Stein Discrepancy) with fewer total iterations. Across all domains, NPC shows superior numerical stability, especially in regions where the solution trajectory exhibits high curvature.

The paper’s contributions are threefold: (1) a unified mathematical view that reveals the common PC structure across diverse homotopy problems; (2) the NPC algorithm that learns predictor and corrector policies via reinforcement learning, eliminating hand‑crafted heuristics; (3) an amortized training strategy that enables a single policy to generalize to unseen instances, validated by extensive empirical results.

Limitations include the reliance on a relatively simple multilayer perceptron for policy representation, which may struggle with very high‑dimensional state spaces, and the need to design domain‑specific reward components. Future work suggested by the authors involves richer architectures (e.g., transformers), multi‑task meta‑learning to further improve cross‑domain transfer, and automatic reward shaping to reduce manual engineering when extending NPC to new homotopy formulations.