Neural Predictor-Corrector: Solving Homotopy Problems with Reinforcement Learning
Authors: Jiayao Mai
Bangyan Liao
Zhenjun Zhao
Yingping Zeng
Haoang Li
Javier Civera
Tailin Wu
Yi Zhou
Peidong Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Homotopy System
Graduated optimization
Reinforcement Learning
Polynomial Equitions System
Gaussian Homotopy
Sampling
Primary area: applications to computer vision, audio, language, and other modalities

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.

Published as a conference paper at ICLR 2026
NEURAL PREDICTOR-CORRECTOR SOLVING HOMO- TOPY PROBLEMS WITH REINFORCEMENT LEARNING
Jiayao Mai Bangyan L Zhenjun Zhao
Yingping Zeng'
Haoang Li
Javier Civera² Tailin Wu? Yi Zhou 1s
Peidong Liu
Hunan University Westlake University University of Zaragoza
'Hong Kong University of Science and Technology (Guangzhou) {maijy, (ereyhhhu@@hmg edu, cn, {l [liaobangyan, liupeidonn ) watesstlakeeeeee.
ABSTRACT
The Homotopy paradigm, a general principle for solving challenging problems, appears across diverse domains such robust optimization, global optimization, polynomial root- root-finding, and sampling. Practical solvers for these problems typi- cally follow a predictor-correctte tics for step sizes and iteration termination, which are often suboptimal and task- specific. To address this, unify these problems under single framework, which enables the design of a general neural solver. Building on this unified view, (NPC), which replaces hand-crafted
(PC) structure, but rely on hand-crafted heuris-
we propose Neural Predictor-Correett heuristics with automatically learned policies. NPC formulates policy selection as sequential decision-makee 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 gener- alizes effectively unseen instances. It consistently outperforms classical and specialized baselines efficiency while demonstrating superior stability across tasks, highlighting the value of unifying homotopy methods into single neural framework.
INTRODUCTION
As a general principle for solving difficult problems, the Homotopy paradigm appears across diverse domains under different names, for example, Graduated Non-Convexity (Yang al., 2020a) and
Gaussian homotopy (Mobahi & Fisher III (015)) for optimization, homotopy continuation (Bates) jet al. (013)) for polynomial root-finding, and annealed Langevin dynamics (Song et al., 2020) for sampling. Specifically, the Homotopy paradigm firstly construct an explicit homotopy interpola- tion from simple, easily solved source problem to complex target problem. Then, the solution of the complex problem is progressively approached by tracing the implicit trajectory along this interpolation path, effectively circumventing the challenges of direct solution.
Practical solvers for these problems often follow predictor-corrector (PC) structure, where pre- dictor advances along the outer homotopy interpolation and a corrector iteratively refines the so- lution (Allgower & Georg, 2012). Despite their widespread use, these solvers rely on manually designed heuristics for step sizes and termination rules, which are typically suboptimal and task- specific. Furthermore, these methods have been independently developed in each domain, and no prior work has systematically unified these efforts under a single framework. We argue that this unification is crucial: it enables the design of general solver that applies across problem instances, rather than requiring ad-hoc, per-problem solutions.
Building this perspective, propose Neural Predictor-Corrector (NPC), plug-and-play framework that replaces heuristic rules with automatically learned policies. Instead of manually
Equual contribution.
Project lead.
"Corresponding authors.

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Published as a conference paper at ICLR 2026
Source Problem
Homotopy
Interpolation
Target
Problem
Optimization
Polynimial Root Finding
Sampling
Figure 1: Homotopy paradigm across domains. The homotopy interpolation (blue loss functions in optimization, green polynomial roots polynomial root-finding, and red probability densities in sampling) is explicitly defined, while the inner solution trajectory (orange curve) must be implicitly tracked.
designed rules, NPC treats the choice of predictor and corrector strategies as a sequential decision- making process (Barto et al. 1989) and employs reinforcement learning (RL) (Kaelbling et al. 1996) to adaptively learn effective policies. Crucially, we adopt an amortized training regime: single offline training phase directly deployed on new instances from the same problem without per- per-instance fine-tuning.
distribution of problem instances produces policy that can be
We evaluate NPC on four representative homotopy tasks: Graduated Kon-Convexity for robust opti- mization (Yang et al. 2020a), Gaussian homotopy for global optimization (Mobahi & Fisher III 2015]. homotopy continuation for polynomial root-finding (Bates et al.) 2013), and annealed Langevin dynamics for sampling (Song et al.) (2000). Through experiments on four representa- tive problems, our approach validated for strong generalization to previously unseen instances. Furthermore, the results reveal dual advantage: our method not only consistently outperforms ex- isting approaches computational efficiency, but also demonstrates superior numerical stability, thereby underscoring the benefits of our proposed architecture.
In summary, our main contributions are as follows:
To the best of our knowledge, we the first unify diverse problems, including robust optimization, global optimization, polynomial system root-finding root-finding, and sampling, under the homotopy paradigm, thereby revealing their common predictor-correttt structure across these problems. This enables unified solver framework, rather than per-problem solutions.
We introduce Neural Predictor-Correetor (NPC), the first reinforcement learning-based framework that automatically learns predictor and corrector policies, replacing hand- crafted heuristics with learned, adaptive strategies.
Extensive experiments across multiple homotopy problems demonstrate that NPC signif- icantly outperforms other methods in efficiency, while achieving higher stability and en- abling efficient, training- free deployment on previously unseen instances.
RELATED WORKS
Although PC solvers appear across multiple domains, these lines of research have largely evolved independently. We review them here and highlight gaps that motivate our work. full discussion of related works is provided in Appendix
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Classical PC algorithms. PC schemes trace solution trajectories along explicit homotopy interpo- lations. In robust optimization, Graduated Con-Convexity (GNC) gradually increases non convexiyy to avoid poor local minima, with iterative solvers performing corrections (Yang al. 2020al Peng
et al. 2023). Gaussian homotopy methods construct progressively less smoothed objectives to track minimizers along bandwidth reduction (Blake & Zisserman 1987] Mobahi & Fisher III| 2015
Iwakiri et al. 2022 Xu (024). Polynomial system root-finding uses homotopy continuation with PC integration to trace roots from simple start system (Bates al., 2013 Breiding & Timme,
2018 Duff' et al., 2019). In sampling, annealed Langevin dynamics and Sequential Monte Carlo de- fine sequences intermediate distributions with PC steps (Song & Ermon 2019 Song et al., [2000]
Doucer et al. 2001). Across all these domains, predictor and corrector components are typically hand-designed, requiring per-instance tuning and limiting generalization.
Learning-based improvements for homotopy workflows. Recent work has introduced learning into homotopy pipelines, showing efficient and effective improvements on Gaussian homotopy (Lin) et al. 2023), sampling (Richter & Berner, 2024), combinatorial optimization (Ichikawa 2024), and polynomial root-finding (Hruby et al.) 2022] Zhang et al.) 2025). However, prior learning-basee
methods either focus single homotopy component require specialized per-instance training.
Reinforcement learning for optimization and sampling. RL. been applied to learn optimizers or adapt algorithmic parameters, showing benefits on some optimization and sampling tasks (Li 2019 Belder al. 2023) Ye et al.) 2025] [Lu et al.] 2025] Yan et al., 2025b Wang et al.) (025). However, these works do not address the full predictor-correcte control problem across diverse
homotopy classes, nor do they leverage amortized training to produce a single policy transferable across instances.
3 HOMOTOPY PARADIGM AS A UNIFIED PERSPECTIVE
In this section, we introduce a unified perspective on diverse problems. We begin in Sec. 3.1 by introducing the homotopy paradigm, general principle that underlies wide range of problems. Next, in Sec. 3.2, we show that the corresponding practical solvers can all be instantiated within a common predictor-corrector (PC) framework. Finally, in Sec. 333] discuss four representative problems together with their homotopy formulations and PC implementations, thereby illustrating the breadth and utility of this unified perspective.
3.1 HOMOTOPY PARADIGM
As shown in Fig. L the homotopy paradigm provides a general principle for solving complex prob- lem g(x). Specifically, the homotopy paradigm defines continuous interpolation H (x, () from a simple source problem H(x,0) 0) f(x) with known solutions to complex target problem H(x, = g(x). By tracing the implicit solution trajectory x x*(() along this interpolation from 0 to 1, one progressively transforms the source solution into the target solution. The source problem and interpolation are explicitly defined by the user, while the target solution is implicitly determined along the trajectory.
varies
3.2 PREDICTOR-CORIS CORRECTOR ALGORITHM
While the homotopy paradigm specifies the abstract principle, an effective algorithm needed to trace the implicit solution trajectory in practice. The PC method (Allgower & Georg] 2012) provides such a concrete algorithmic framework. As shown in Fig. PC decomposes trajectory tracking into two complementary steps:
Predictor: Determines the next level of the homotopy interpolation and predicts the solu-
tion's position at that level.
Corrector: Iteratively refines the predicted solution align it with the true solution tra-
jectory, thereby preventing the accumulation of bias across levels.
The choice of predictor level schedule and corrector iteration count is often heuristic. Suboptimal settings can lead to inefficiency, instability, or failure to follow the trajectory accurately, motivating the development of adaptive or tearning-based strategies for robust and efficient solution tracking.
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Predict
Predict
Correct
Correct
Correct
Predict
Target
Homotopy
Source Problem
Problem
Interpolation
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 back onto the solution trajectory. Orange curve denotes the implicit solution trajectory, as in Fig.
Figure 2:
3.3 REPRESENTATIVE HOMOTOPY PROBLEMS AND PRACTICAL SOLVERS
To illustrate the breadth of homotopy paradigm applications, we descrihe four representative prob- lems together with their corresponding homotopy interpolations and PC implementations.
1) Robust Optimization (Graduated Non-Convexity, GNC): Robust loss functions (e.g., Ge- man-McClure (Black & Rangarajan, 1996)) mitigate the effect of outliers. However, they intro- duce strong non-convexity, increasing the risk of poor local minima. Graduated Non-Convexity (GNC) Yang et al. 2020a) addresses this challenge by defining a homotopy interpolation:
H(x, txx
2 v:)a
where é is predefined parameter that controls the robustness of the GM loss, r(::)) represents the residual function, and denotes the measurements. This interpolation smoothly transitions from convex quadratic loss (H (x, 0) Ei=1 r(x, ²) the original non-convex Geman-MeClure loss r(x.;))
(H(, g(x) E== (x+r(x.)))) The predictor gradually increases non-convexity according predefined schedule, while the corrector refines the solution at each stage, often non-linear least squares optimizer (e.g.. Lovenberg-Marquarde strategy has proven highly effective in problems such as point cloud registration under severe outlier contamination (Yang et al. (020b). Details are provided in Appendix A.1
Marquardt algorithm (Levenberg] 1944)). This homotopy
2) Global Optimization (Gaussian Homotopy, GH); Many optimization problems suffer from
highly non-convex landscapes with narrow basins of attraction, making it difficult for solvers to converge global or high-quality local minima. [wakiri al. (2022) address this challenge by progressively smoothing the target function through convolution with Gaussian kernel (N0,, to?):
H(x, t) = g(x) NO tO))
(2)
where * denotes the convolution operator. This Gaussian smoothing enlarges the basin of attraction, allowing solvers to approach promising regions more reliably. The predictor progressively reduces the kernel bandwidth, while the corrector refines the solution each stage. Details are provided in Appendix A.2
3) Polynomial Root-Finding (Homotopy Continuation, HC): Root-finding for polynomial sys- tems is challenging due to multiple solutions and computational complexity. [Bates et al. (2013) address this by starting from source system f(x) = with known roots and defining a linear homotopy:
= = (1-t)f(x) + tg(x).,
(3)
tracing the solution trajectory from the source the next solution along this path, while the corrector refines it using Gauss-Newton (Björck 2024) iteration at each step. ensuring accuracy along the trajectory. Details are provided in Appendix A.3
the target roots. The predictor extrapolates
4) Sampling (Annealed Langevin Dynamics, ALD): Sampling from complex, digh-dimessional distributions challenging due to multi-modality and slow mixing. [Song al. (202)) address this
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External Action: Predictor's Next Level
Internal Action:
Corrector's Tolerance
NPC
Environment
Reward: Accuracy + Efficiency Bonus
1. Predictor's Homotopy Level Corrector Statistics (Tolerance, Iteration)
State:
Convergence Velocity
Figure 3: RL formulation of the proposed Neural
(NPC). each homo- topy 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 toler- ance, and receives rewards designed to balance accuracy and efficiency.
by constructing a homotopy between simple source distribution (e.g., Gaussian) and the target distribution:
H(x, I) x exp ( (1)
(4)
The predictor schedules the intermediate distributions, while Langevin dynamics acts the correc- tor at each step, iteratively refining samples to match the current intermediate distribution. Details provided in Appendix A.4
These examples collectively highlight the broad applicability of homotopy paradigm and the central role of predictor-corrector strategies, motivating the need for learning- baasedg-basedd policy optimization.
NEURAL PREDICTOR PREDICTOR-CORRETTOR CORRECTOR WITH REINFORCEMENT LEARNING
This section introduces the Neural homotopy problems that replaces heuristic step-size and termination rules with neural parameter- izations learned via RL. As shown in Fig. NPC reformulates the predictor-corrector process as a sequential decision problem: the predictor advances the homotopy level, while the corrector en- sures accuracy, both guided by adaptive policies. We first present the NPC formulation (Sec. 4.1. followed by its training with reinforcement learning (Sec. 4.2
(NPC) framework, general approach for
4.1 NEURAL PREDICTOR CORCCCTTT-O
Classical PC algorithms differ across homotopy problems in how they define prediction and cor- rection, yet share key limitation: their step-size schedules and termination criteria are governed by fixed heuristics. Such heuristics fail to adapt to varying solution trajectories, where small steps needed for sharp transitions but larger steps improve efficiency when the trajectory is smooth.
The NPC addresses this limitation by parameterizing the decision rules with neural network (NN). Instead of hand-crafted heuristics, NPC learns flexible and adaptive strate-
Algorithm Neural Predictor-Correcte Solver
Input: Homotopy problem H
1: Warm up for initialization.
2: while f. do
gies that generalize across problem
3:
NPC: (Ata, én or <.Th NN(t)-1, 6n
in- Tn-11) Tn-1)
instances. The entire PC process is modeled Markov Decision Pro- cess (MDP), in which, at each ho- motopy interpolation level, an agent observes the state and selects ac- tions that govern the procedure.
4: Predictor: Update interpolation level tn = tn-1 + Atn
5: Predictor: Predict xt,, al level f,,
6: while [[ ft,) en and quax do
7: Corrector: Perform one step correction end while
8:
9: Collect corrector statistics e., in
10: Collect convergence velocity
The state encodes both progress
11:
end while
and dynamics:
Output: Optimal solution xf_1
Homotopy Level: Current position along the interpolation path.
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Corrector Statisties: Iteration count and attained tolerance from the previous step, captur- ing both convergence efficiency and deviation from the predicted trajectory.
Convergence Velocity: Relative change in an optimality metric between consecutive lev- els, reflecting the speed of convergence. For optimization and root-finding, this is the relative change in the objective value. For sampling, it is the change in statistical dis- tance such as Kernelized Stein Discrepancy (KSD) (Liu et al., 2016) between the empirical sample distribution and the target distribution across consecutive levels.
Given the state NPC outputs two-part action a:
Step Size At: Controls the predictor's ' advance along the homotopy path.
Corrector Termination: Convergence threshold or maximum number of updates, bal- ancing accuracy and efficiency.
As shown in Algorithm [ the NPC solver operates in an iterative loop trace the solution path of given homotopy problem H. Fach iteration consists of three key stages. First, a neural network (the NPC module) dynamically determines next actions for both the predictor and corrector. Second, the predictor ad advances the homotopy level to fn and predicts the solution X,, at this level. Third, the cor- rector iteratively refines this prediction until the convergence criteria are met. Finally, performance statistics are collected and fed back to the NPC module to inform its decisions in the next iteration, creating an adaptive, closed-loop system.
4.2 REINFORCEMENT LEARNING FOR NPC
Because the predictor-corrector procedure is non-differentiable and early decisions influence the en- tire trajectory. supervised or self-supervised training is inadequate. These approaches would require assuming that local geometric structures of the solution trajectory remain consistent across instances, which rarely holds in practice. We instead employ RL, which inherently evaluates sequential deci- sions by their cumulative effect and enables learning policies that generalize across instances within the same problem class. The reward function is designed to promote both accuracy and efficiency:
Step-wise Accuracy (r²²); Encourages faithful trajectory tracking, based on convergence
velocity or relative error change in the target problem.
Terminal Efficiency Bonus ():*T)); Rewards shorter corrector sequences, formulated as Thax - T, where I max is predefined upper bound and T is the total corrector iterations.
Consequently, the cumulative reward R for an episode is defined as: R (E A + Agr="! where Å1 A2 are scaling coefficients detailed in Appendix A This reward design enables agent to balance accuracy and efficiency across the homotopy trajectory.
Remarks amortized training for generalization. Sequential decision-making in homotopy problems entails that early step-size choices affect all subsequent levels. Self-supervised learning fails in this context because measuring the future contribution of step size is infeasible: it depends on the local geometric properties of the trajectory at future homotopy levels, which are unknown in advance. Relying such assumptions risks overfitting the training landscapes. This challenge is analogous to the dilemma discussed in (Li 2019), where although the problem domains differ, the core issue of long-term dependency and overfitting is similar.
Reinforcement learning, by contrast, inherently evaluates actions based on cumulative outcomes, allowing NPC adapt to diverse solution trajectories without assuming consistent local geome- try. Amortized training further improves generalization: by training over a distribution of problem instances, NPC learns a policy that can be applied efficiently unseen instances within the same problem class.
5 EXPERIMENTS
5.1 IMPLEMENTATION DETAILS
Following the RL formulation in Sec. we employ Proximal Policy Optimization (PPO) (Schulman
et al. 2017), on-policy algorithm well-suited for continuous state and action spaces. Implemen-

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Table 1: Performance on the GNC point cloud registration task. Rotation and translation emror (En and F2) are reported on 1og10 scale.
Table 2: Performance on the GNC multi- view triangulation task. Reconstructed 3D point errors (LEp) are reported on a log10 scale.
Sequence
Method
log( (R)
log( (..)
lter
Time
Sequence
Method
1, +
Iter
Time
Classic GNC
-0.85
-2.66
783 161.00
Classic GNC
-4.62
142
81.98
bunny
IRLS GNC
-0.85
-2.75
309 61.59
reichstag
IRLS GNC
1.74
37
10.72
Ours OUrC +GNC
-0.85
-2.71
169
19.15
Ours' +GNC
-4.72
21
14.18
Classic GNC
-1.12
-2.89
486
89.34
Classic GNC
-5.15
195
91.23
cube
IRLS GNC
-1.10
-2.90
141
26.13
coeur IRLS GNC
0.50
16
21.31
Ours +GNC
-1.11
-2.86
86
7.86
Ours' +GNC
-4.84
20
14.14
Classic GNC
-0.80
-2.82
859
177.11
Classic GNC
-4.81
136
80.50
dragon
IRLS GNC
-0.80
-2.82
486
95.93
st. st.pt.q
IRLS GNC
1.00
19
27.92
Ours +GNC
-0.80
-2.80
201
26.42
Ours' +GNC
-4.98
18
15.55
The agent is trained on the Aquarius sequence for the point cloud registration task.
The agent trained on the Aquarius sequence
for the point cloud registration task.
tation is based on the open-source Stable Baselines3 library (Raffin et al. 2021). The policy and value functions are parameterized multi-layer perceptrons (MLPs) with two hidden layers of 16 units each and ReLU activations. All other hyperparameters use the default values provided by Sta- ble Basclines3. To account for varying problem formulations and noise levels across tasks, reward signals are scaled appropriately to ensure stable learning and comparability across tasks. Details conducted
provided in Appendix [A] Áll experiments 12700KF CPU and an NVIDIA GeForce RTX 3060 GPU, unless otherwise specified.
12-core 5.0 GHz Intel Core i7-
In all tables, Iter 1 records the total number of corrector iterations (rather than predictor iterations, which are more commonly used to measure progress in homotopy problems), and Time + reports runtime milliseconds. The best results are bolded and the second-best results in Tab. 3 are under- lined. All results represent the average over 50 independent trials.
5.2 PROBLEM 1: ROBUST OPTIMIZATION VIA GNC
We evaluate NPC in the context of robust optimization using the GNC framework, comparing against the classical GNC (Classic GNC) approach and the iteratively reweighted least-squares (IRLS) version (Peng et al. 2023). The evaluation covers two spatial perception tasks with high outlier ratios: point cloud registration (Alexiou et al., 2018) (95% outliers) and multi-view triangu- lation (Jin et al.) 2021) (50% outliers). Our NPC model is trained solely on the Aquarius dataset from the EPFL Geometric Computing Laboratory, demonstrating its cross-instance generalization capabilities.
report the rotation error (En) and translation error (E,) in Tab. [ for each method. Additionally, Tab. presents the 3D point recon- struction error (Ep), defined as the Euclidean distance between reconstructed and ground-truth 3D points. As shown in Tabs. I and 2 IRLS, tailored for a specific task, performs poorly on triangulation and lacks generalization. In terms of efficiency, NPC significantly boosts GNC's performance: on point cloud registration, re- duces iterations by approximately 70-80% and runtime by 80-90% without compromising accuracy. These results demonstrate that NPC preserves the robustness of Classic GNC while substantially improving efficiency and generalization.
Following the metries defined in (Yang & Carlone 2019).
NPC achieves accuracy comparable to Classic GNC, whereas
5.3 PROBLEM 2: GLOBAL OPTIMIZATION VIA GH
We evaluate NPC in the GH setting for non-convex function minimization. We compare our method
with two categories of baselines: (i) the single loop GH methods, SLGH, (y = 0.995) and SLGH, (1) 10-4) (Iwakiri al. 2022), (ii) the Gaussian smoothing method, PGS (N 20) (Xu 2024).

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and (iii) the learning- based method, CPL (Lin jct al., 2023). Performance is evaluated on three 2-dimension non-convex bench- marks: the Ackley (Ackley) 1987). Himmel- blau (Himmelblau et al.) (172), and Rastri- gin (Rastrigin 1974) functions. The optimal value f(x") is 0 for all problems.
Table 3: Performance on GH non-convex func-
tion minimization benchmarks.
Problems
Method
f(x")
Iter
Time
Classic GH
0.07
501
16.25
SLGH,
0.12
1839
56,71
SLGHA,
0.26
568
28.45
2d Ackley
As summarized in Tab. 3] NPC-accelerated GH achieves substantial reduction in it- erations and runtime compared to Classic GH, while maintaining comparable solution quality, SLGHa and PGS occasionally fail to reach the optimum, especially on Him- melblau, highlighting the challenge these landscapes pose for exee-scheduue homotopy methods. CPL is designed to learn the solu- tion path for a specific, fixed-coefficient prob- lem instance. Consequently, training time must be factored into the runtime, negat- ing any efficiency advantage. Overall, these results show that NPC provides an notable trade-off between efficiency and robustness. generalizes well to unseen problem instances while accelerating convergence.
PGS
0.07
200
14.32
CPL
0.01
1701.61
Ours*+GH
0.05
359
12.31
Classic GH
0.00
501
11.39
SLGH,,
0.00
1839
41.70
SLGHz
2.57
75
2.57
Himmelblau
PGS
1.18
200
11.33
CPL
0.00
2160.17
Ours2+GH
0.00
345
8.91
Classic GH
0.00
501
23.76
SLGH,
0.00
1839
78.21
SLGH,
0.34
319
19.64
Rastrigin
PGS
0.14
200
11.94
CPI.
0.57
790.38
Ours2+GH
0.00
247
11.84
2 The agent trained on the Ackley functions with randomized parameters and evaluated on the canonical fixed-parameter version.
5.4 PROBLEM POLYNOMIAL ROOT-FINDING VIA HC
We evaluate NPC in the context of polynomial system root-finding using HC. Experiments are con- ducted on two categories of tasks: polynomial system benchmarks KKttuura 1990 Himmelblau al., 1972; Rastrigin| 1974) and computer vision problem (UPnP (Kneip al. (2144) for gen-
eralized camera pose estimation from 2D-3D correspondences. Tab. [ lists the specific polynomial systems used, with the first entries as classical benchmarks and the last as computer vision task. We compare NPC-accelerated HC with Classic HC and Simulator HC (Zhang et al. 2025). Both Classic
HC and NPC-accelerated HC use the monodromy module in Macaulay2 to generate start systems following (Duff) 2021). while Simulator HC pre- trains a regression neural network to predict the start system, relying on physical modeling of each problem. Consequently, Simulator HC is inapplicable standard polynomial benchmarks. The NPC agent is trained on polynomial systems from the 4-view triangulation task with random- learn generalizable policies. As shown in Tab. NPC consistently tracks all target solutions successfully while reducing the number of iterations and runtime compared to Classic HC. Notably, Simulator HC relies on task-specific pre-trained network, which limits its generality, and its runtime is not directly compa- rable since it is implemented in C++. In con- trast, NPC provides general-purpose, adaptive solver that achieves accelerated convergence with- out tar-task pre-training.
Table Performance on HC polynomial sys- tem benchmarks. Suce. denotes the success rate of tracking to root, and Time reports the average tracking time per solution path.
Problems
Method
Suce.
Iter
Time
Classic HC
100%
39
2.22
katsura10
Ours
100%
7
0.65
ized coefficients
cyclic7
Classic HC
100%
41
1.96
Ours
100%
8
0.64
Classic HC
100%
53
8.25
UPnP
Simulator HC
100%
100
Ours"+11e
100%
29
3.86
:: Runtimes are not directly comparable, as Sim- ulator HC is implemented in C++, while the other methods are in Python.
The agent trained polynomial systems from the 4-view triangulation task with randomized coef- ficients.
5.5 PROBLEM SAMPLING VIA ANNEALED LANGEVIN DYNAMICS (ALD)
We evaluate NPC in the context of ALD for sampling from complex distributions. Target dis- tributions include a 40-mode Gaussian mixture model (GMM), a 10-dimensional funnel distribu- tion Neal 2003), and 4-partiele double-well (DW-4) potential (Köhler al. 2020). The NPC
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agent is trained the 10-mode GMM with randomly sampled coefficients to learn generalizable policies for accelerating ALD. We compare our method against classic ALD (Song et al. 2020) and, where applicable, iDEM (Akhound- Sadegh et al. 2024) for GMM and DW- with 10 saved samples. Evaluation metrics are the Wasserstein-2 distance (W2) (Peyré et al.) 2019) and the Ker- nelized Stein Discrepancy (KSD) (Liu et al. 2016). As shown in Tab. 5 NPC-accelerated ALD requires signifi- cantly fewer iterations while achieving W/ and KSD values comparable to clas- sical ALD. Although iDEM attains lower W2 some tasks, it relies extensive per-task computation and not directly comparable in runtime. Overall, these results demonstrate that NPC effectively accelerates sampling while maintaining high-quality approximations of the target distributions.
Table 5: Performance on ALD sampling. Wasserstein-2 distance (W2) and Kernelized Stein
Discrepancy (KSD) evaluate sample quality.
Distributions
Method
W2 KSD 4 Iter
Time
Classic ALD
11.57
0.0037
410
1353.16
40-mode GMM
iDEM
7,42 0.0037
1000
Ours' +ALD 11,91 0.0040
110
772.34
Classic ALD 30.91 0.0382
410
754,48
funnel (d=10)
Ours"+ALD 31.02 0.0343
105
686.55
Classic ALD
3.77 0.0911
410
1337.70
DW-4
iDEM
2.13 0.0911
1000
Ours*+ALD +ALD
3.47 0.0899 105
711.66
Runtimes not directly comparable, iDEM is mea- sured
powerful NVIDIA RTX A6000 GPU. The agent is trained on the 10-mode GMM with randomly sampled coefficients.
5.6 ABLATION STUDY OF RL STATE COMPONENTS
To assess the contribution of each component in the RL state, perform ablation study the six datasets used for the GNC point cloud registration task, retraining the NPC agent with one component removed
time. As summarized in Tab. 6 removing any single state component causes the agent to adopt a more conservative strategy, resulting in an increased number of corrector iterations relative to the full state. This tendency typically manifests the agent selecting smaller predictor step sizes or stricter corrector tolerances to ensure convergence in the absence of complete information. This indicates that each state component, i.e., homotopy level, corrector tolerance, corrector iteration count, and convergence velocity, pro- vides essential information for efficiently guiding the homotopy solver. Notably, the results suggest that corrector statis- ties (iee,, corrector tolerance and itera- tion) are the most informative parts of the state, as their removal leads to the largest performance drop.
Table 6: Effect of each RL state component on NPC performance.
Homotopy Level
Corrector's Corrector's Convergence Velocity
Alter
Tolerance
Iteration
+21
X
+64
V
+52
+38
5.7 ANALYSIS OF EFFICIENCY-PRECIIS TRADE-OFF
We analyze the efficiency-preccsioo trade-off by benchmarking our NPC-accelerated method against classical GNC and ALD. Classical approaches require manual tuning of homotopy parameters, re- sulting performance curve where higher precision typically demands more iterations. By con- trast, our NPC-accelerated method bypasses this manual exploration by learning a policy that di- rectly identifies an optimal operating point. This learned policy inherently balances the predictor step size and corrector tolerance maximize efficiency benefit visualized in Fig. 4 For both GNC and ALD tasks, the single point representing our method lies well below the classical trade-off curves, clearly illustrating substantial reduction in iterations at comparable precision.
given precision level. The practical
6 CONCLUSION
This paper introduces Neural Predictor-Correcct (NPC), reinforcement learning framework for homotopy solvers. By unifying diverse problems, including robust optimization, global optimiza- tion, polynomial system root-finding, and sampling, under the homotopy paradigm, their solvers

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1000
500
NPC GNC
NPC ALD
800
Fitted
Iterations
600
Iterations
300
200
400
200
10
10
10
10
10
10 92
10
11
12
15
16
17
Rotation (Deg)
Wasserstein-2 Distance
(a) GNC point cloud registration.
(b) ALD sampling.
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.
are shown to universally follow a PC structure. NPC replaces handcrafted heuristics with adaptive learned policies and employs an amortized training regime, enabling one-time offline training and efficient, training-free deployment on new instances. Extensive experiments demonstrate that NPC generalizes effectively to unseen instances, consistently outperforms existing approaches in compu- tational efficiency, and exhibits superior numerical stability. These findings position learning-baat based policy search practical, generalizable, and efficient alternative traditional heuristic strategies. Looking ahead, this paradigm opens promising avenues for extending homotopy methods to broader classes of optimization and sampling problems. Nonetheless, also acknowledge its current limi- tation, which is discussed Appendix
7 ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (62202389, 625B2151, 62427813, 62403401). Spanish grants PID2021-127685NB-I00 and PID2024- 155886NB-100, Aragón grant 23R, the Al Research and Learning Base of Urban Culture (2023WZJD008), and the Westlake University-Muyun Joint Research Institute, the Westlake Edu- cation Foundation.
8 ETHICS STATEMENT
Our work unifies diverse problem domains governed by the homotopy paradigm into a single frame- work and, based on proposes general, learning-based solver NPC. Our experiments are con- ducted on publicly available academic benchmarks and synthetic data, involving no human subjects or sensitive personal information. We do not foresee any direct negative societal impacts dual-use concerns, as the primary application of our work is to provide more efficient and robust tool for scientific inquiry.
REPRODUCIBILITY STATEMENT
To ensure reproducibility, we specify the sources for all real-world datasets and the parameters used to generate synthetic data. In addition, Appendix A provides additional implementation details, covering the specifie problem formulations and the hyperparameters used in our experiments. Our code and pretrained models will also be released to the public.
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A
IMPLEMENTATION DETAILS
A.1 DETAILS OF PROBLEM ROBUST OPTIMIZATION VIA GNC SEC. 5.2
A.1.1 THE GRADUATED CON-EONVY ALGORITHM
Optimization problems that can formulated as least-squares can utilize the robust kernel from Eq. (T. which is represented as:
min H(x,)) t).
(5)
xEX,ET
The GNC algorithm utilizes Black Rangaraan Duality (Black & Rangarajan 1996) to reformulate
Eq. 55 into:
x min XEX
war (y. + THH (w:)
where wi the weight of the 8th measurement y:, and the function + 9 (-), whose expression depends the choice of the robust cost function H,, defines penalty on the weight wi. When He defined by Eq. (I). & H. (w,) defined by HH, (w.) = ( - 1)? Moreover, the weight can a function of only and residual
be solved in closed form form
Predictor Reformulating the problem as Eq. 6 simplifies the prediction step to updating the each weight wa using Eq. (7). rather than predicting the optimization variable x.
2
Wi =
1T2 (x)
Corrector We correct using nonlinear optimization method defined by Eq. (8)
min XEX
war (y:.x)
(8)
In our experiments, point cloud registration employs a Kauss-Newton corrector, while multi-view triangulation uses more robust
(LM) algorithm.
Details of experiment. For the point cloud registration task, the reward scaling
A 10s
and A2 = 10-3, For the multi-view triangulation task, the reward scaling is set to à 10-1 and A2 10 due its noise scale being significantly larger than that of point cloud registration.
A.2 DETAILS PROBLEM GLOBAL OPTIMIZATION VIA GH (SEC. 5.3
A.2.1 THE GAUSSIAN HOMOTOPY ALGORITIIM
The equivalent expression for Eq. is given by:
H(x, 1) = g(x+|* = Eor- +(+)))+++)))
(9)
where k(()) (2x) $ -ل61" is referred to the kernel.
Predictor. The prediction process in Gaussian homotopy implicit, we only modify the shape of (x,t) by varying the predictor's homotopy level f.
Corrector: The correction for is performed using momentum method (Polyak 1964), with the gradient update formulated as
V V Hxx, Sv:
(10)
X
ov++1
where V, is the velocity vector, with the initial velocity Vo set to the zero vector, is momentum coefficients, controlling the influence of past gradients, and is the learning rate, which determines the step size of the update. We set a 0.01 and 3 = 0.8 in our experiment. As the analytical computation of the gradient V H (x,, f) is not feasible for some Gaussian homotopy functions,
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employ zeroth-order method to obtain a numerical approximation. The calculation formula is as follows (Nesterov Spokoiny 2017):
(xH(xx,t) = Vx VaBo-woonllxx
1(9(x + - * +o] (11)
Details of experiment. The reward scaling is set A and X 10-3
A.2.2
THE NON-CONVEX FUNCTION MINIMIZATION BENCHMARKS
Ackley Optimization Problem (n-dimensions)
Ackley 1987):
-.2 f(x) = exp
2
exp
cos (os()));))
+ +
(12)
n
1=1
Himmelblau Optimization Problem (Himmelblau et al. 1972):
f(z,y) y) (x +y 11)2 + (a 5² -))2
(13)
Rastrigin Optimization Problem |Rastrigin 1974):
f(x v) = 10 x + x² - 9 cos(22xx) - cos((Tr)
(14)
A.3
DETAILS OF PROBLEM 3: POLYNOMIAL ROOT-FINDING VIA HC (SEC. 5.4
A.3.1
THE COMOTOPY-CONTINUNN ALGORITHM
The polynomial system root-finding problem modeled in the form Eq.
Predictor.
The prediction of x(t + At) is performed using the Padé approximation. The Padé
approximation polynomial Rn,m (I)
R, (x) has the following form:
Qm (x)
m (I) Po pi
Puit
(15)
1 91 + + ama
1+ Σk=1 Cka
It is equivalent to the power series given in Eq. (T)..
(xx)
C
(16)
The basic Padé approximation principle is that, given two integers m,n n E NN {0), we can find two polynomials P, (z) of degree most and (m) of degree at most m, such that the differ- ence @m (x) - (x) has an order of approximation of at least n + m + 1. In fact, this is mathematically equivalent the requirement:
(z - (x) amz++))))
(17)
where Oz denotes power series of the form EnEN G,æ"..
In our implementation, we n 2 and we can derive the following coefficients based on
Eq. (T7
C3 91
02
= Co>
(18)
Pi 01 + q Co:
= + C1.
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where cu x(t). G x(t), 0 (t),
c3 (t).
We obtain the derivative of x by
differentiating ()) with respect to
0H (1)
0H
0x
a1.
0H (t)
8 H
8
((t)
(19)
dx
Oxiit
dt?
0H (1)
H²
H
0'H
(1)
x1) +
Oxöt
oxot
013
Consequently, x(1 + At) is according the following equation:
Po + P1 At + P2
x(t + At)
(20)
1 1+ At
the denominator in Eq. (0 approaches zero, we revert to using a power series to predict (I++At). In this case, the prediction takes the form x(t At)
+ c Al cz cz
Corrector. We employ a Newton corrector in our experimental setup. At each iteration, x is updated according to the following equation until the convergence criterion Ax < 6 is met.
OH [x, + Al) Ax H(x, L A!),
8x
(21)
x x Ax.
Details of experiment. The reward scaling is set ò 10-3 and X2 10-1
A.3.2
THE POLYNOMIAL SYSTEM BENCIIMRRSS
The Katsura-n Polynomial System
Katsura 1990):
fo
Σ Øj
-1=0
(22)
fk+1: In +
Fj
Ig =0
(for 1e 0, 0,1, ......n 1)
n+1
The Cyclic-n Polynomial System (Davenpor 1987):
fo
3z =
A:
Tff(j+1) (mod n) 0
1=0
(23)
n-1 2
fn-2:
((j+H)
(mod nj
= 0
==0
k=0
fa-1:
Tjj
11
The Noon-n Polynomial System Noonburg 1989):
cr, + +1 0 for = 1,...,n
(24)
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In our implementation, set 1.1.
The Chandra-n Polynomial System (Kelley 1980):
2TTZ CFK 1
+
2n = 0 for k = 1,... .....
(25)
k
i=1
In our implementation, we 0.51234.
A.4 DETAILS OF PROBLEM 4: SAMPLING VIA ALD (SEC. 5.5
A.4.1 THE ANNRALED LANGEVIN DYNAMIC SAMPLING ALGORITHM
Annealed Langevin dynamics sampling obtains initial sample points from simple distribution and uses series dependennd potentials to control the update of the samples, as shown Eq. (4). Let the time-dependent potentials be H H(x, I) exp ( (- t))(x) (g()))
Predictor. The prediction process in ALD sampling is implicit, as we only modify the shape of H (x,,)) t) by varying the predictor's 's homotopy level t.
Corrector. each iteration of the corrector, the positions of the samples are updated using the following formula:
x {xHH(x,))) + Vea,
(26)
where & is a Langevin step size, and _N0 La) is Gaussian noise vector.
Details of experiment. The reward scaling is set A 10, and A2 10-3.
A.4.2 DISTRIBUTIONS
The 10-dimensional funnel distribution.
The 10 dimensions Funnel distribution defined as
3o NN0, o²) 23/200 NN(0, e²),
(27)
for i =1,-... 1,...,9 9.
The funnel potential given as
30
O(x) = 20
x
(28)
2 Y=1
The 4-particle double-well (DW-4) potential. The DW-4 potential defined as
g(x)
2r Σ a(dy do) b(d,, do)² c(di do)
(29)
where dis ||=; xz the Euclidean distance between particles and j. In our implementation, we set 7 1, 0, -4, and = 0.9.
A.4.3 METRIC
Wasserstein-2 distance (W2). The Wasserstein-2 distance (Peyré et al. 2019) is given by
$
W2(pv)) = iin / (z., (xx(())))))))))))))))) y) dædy
(30)
where и is the transport plan with marginals constrained and respectively. In implemen- tation, we we use the Python Optimal Transport (POT) package (Flamary et al., (021) to compute
this metric.
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Kernelized Stein Discrepancy (KSD). The Kernelized Stein Discrepancy (Liu et al. 2016 is defined as
uy( sT(c) (oe)) k(r, x' (og(=') s4(x) Vx,
+ Vak(a, x') " sa(x') + trace Va. S(p. q) -' (2, x')]
(31)
where sy -V xx), and k(æ, x²) is positive definite kernel. Specifically, we use the standard RBF kernel for KSD computation in this work.
B BACKGROUND ON REINFORCEMENT LEARNING
Reinforcement learning (RL) (Kaelbling et al. 1996) provides a natural framework for learning adaptive strategies. It formulates sequential decision-making as Markov Decision Process (MDP) with state space S, action space A, transition dynamics p(&(+1 11 [& 0z), initial state distribution (o())). reward function r(s;, oe). and discount factor y (0, 1]. The goal is find an optimal policy π S - A that maximizes the expected cumulative reward along a trajectory (so, , ST):
E
y 1 (s,, a1)
(32)
C FULL RELATED WORK
Although mentioned in previous sections that methods from different fields essentially share the same predictor-corrector spirit, they have long evolved independently of each other. Our work the first to unify these methods. In this section, we will review 1) classical predictor-corrector methods; 2) learning-base improvements on predictor-correcto methods; 3) efficient optimization and sampling methods via reinforcement learning.
Classical PC algorithms. 1) Robust optimization: A core related technique is Graduated Non- Convexity (GNC), first proposed by (Yang al.. 2020a). This method employs predictor-correcto approach with non-linear least-squares solvers to compute robust solutions. However, it relies on hand-crafted, fixed iteration schedule, making it unsuitable for real-time robotics applications. Building on this work, [Png et al.] (2023) established a connection between GNC and the iteratively reweighted least-squares (IRLS) framework, based on which they designed novel iteration strategy that achieved faster speeds in point cloud registration tasks (Liu et al. 2023 Yan et al.) 2025a Chen ct al., 2025a Liao et al. 2024). Nevertheless, this strategy's 's lack of generalizability to other prob- lems remains its primary limitation. 2) Gaussian homotopy optimization: The underlying principle of this area was first introduced in (Blake & Zisserman, 1987). More recently, [wakiri et al. (2022) proposed novel single-loop framework for the Gaussian homotopy method that simultaneously performs prediction and correction. Subsequently, Xu (2024) improved the algorithm's 's convergence rate by adding exponential power-N transformation prior to the Gaussian homotopy process. 3) Polynomial root-finding: Homotopy continuation (Bates et al. 2013), a numerical method for finding the roots of polynomial systems, uses a predictor-correcne
scheme to track solution paths. Subsequent methods by (Breiding & Timme 2018) and (Duff et al. (2019) analyzed the properties of polynomials introduce various improvements, enhancing the algorithm's 's speed. 4) Sampling: In generative modeling, annealed Langevin dynamics (Song & Ermon 2019: Song et al. 2020) utilizes a predictor-correcte method to sample from image probability distributions, where the correction step uses Langevin dynamics to restore samples
equilibrium state. Similarly, Sequential Monte Carlo (SMC) methods (Doucet et al., (00T)) also apply predictor-corretto approach to sample from posterior probability distributions, with correction step that employs importance sampling weight the samples.
Learning- based improvements for homotopy workflows. 1) Gaussian homotopy optimization: Lin et al. (202)) is a novel model-based method that learns the entire continuation path for Gaus- sian homotopy. However, this approach requires specialized training for each problem. 2) Polyno- mial root- noot-finding: Focusing on the more specific sub-problem. of polynomial system root-finding, both (Hruby et al. 2022) and (Zhang al. 2025) propose learning-based methods to determine
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the optimal starting system for homotopy continuation. 3) Combinatorial optimization: Ichikawa (2024) proposes the Continuous Relaxation Annealing strategy, aiming to enhance unsupervised learning solvers for combinatorial optimization problems. 4) Sampling: Richter & Berner (2024)
establishes a unifying framework based on path space measures and time-reversals, and proposes a novel logvvariance loss that avoids differentiation through the SDD solver.
Reinforcement learning for optimization and sampling. 1) Optimization: Li (019)) proposes a general framework by formulating optimization algorithm reinforcement learning problem, where the optimizer is represented as a policy that learns to generate update steps directly, aim- ing to converge faster and find better optima than hand-engineered method. [Bllder al. (2023) utilizes reinforcement learning (Chen et al., (2025b) to train an agent that dynamically selects the damping factor in the Levenberg-Marquaree algorithm (Levenberg) 1944) accelerate convergence by reducing the number of iterations. 2) Sampling: Ye et al. (2025) employs reinforcement learn- ing to adaptively predict the denoising schedule via optimizing a reward function that encourages high image quality while penalizing excessive number of denoising steps. Wang et al. (2025) proposes general framework named Reinforcement Learning SurrooliisHHattnng which aims to automatically design and optimize Markov Chain Monte Carlo (MCMC) samplers.
D LIMITATION AND FUTURE WORK
One limitation of our work that the NPC agent's ss reward scale currently requires manual tuning for each problem instance based its noise level to ensure stable and efficient training. The scale of step-wise rewards influences the training process's 's convergence time, while an oversized terminal reward can nullify the guidance from step-wise rewards. This imbalance prevent the agent from correctly tracking the solution trajectory, causing it to adopt myopic strategies to prematurely reach terminal state. We conduct experiments the point cloud registration task with different reward scaling factors. The results are shown in Tab,
Table 7: Comparison of results under different reward scaling settings. Convergence Steps (Training) denotes the approximate step count where the cumulative reward stabilizes during train- ing.
Convergence Steps (Training)
Method
Reward Scaling
log(ER) 4
log( E,) 4
Iter
A 103, A2 10-3 (*)
3M
-1.11
-2.86
86
A1 = 102 A2 = 10 -
2M
-1.08
-2.67
70
Ours+GNC
A1 103, A2 10
6M
-1.08
-2.91
74
A 10", A2 10-2
Fail
Classic GNC
-1.12
-2.89
486
IRLS GNC
-1.10
-2.90
141
(+): The settings used in the paper.
To address this, two avenues for future work are promising. The first to develop a mechanism that automatically adapts the reward scale. A more fundamental solution would be to investigate adaptive normalization techniques for the reward function, making the learning process inherently robust and eliminating manual tuning.
E FULL EXPERIMENTAL RESULTS
This section provides complete experimental results, which are summarized in Secs, 5.2 to 5 due to limited space. Tab. g shows the full results for the point cloud registration experiments via GNC, Tab. 9 shows the full results for the non-convex function minimization experiments via GH, and Tab. 10 presents the detailed results for the root-finding experiments polynomial systems via HC. In addition, we present box plots for a subset of the experimental results in Fig. to visually compare the different methods.
19

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Published as a conference paper at ICLR 2026
Table 8: Performance the GNC point cloud registration task. Rotation and translation errors RER and E,) are reported on log10 scale.
Sequence
Method
log( Ex)
log(Ee)
Time
Classic GNC
-0.85
-2.76
161.00
bunny
IRIS GNC
-0.85
-2.75
61.59
Ours' Ours++GNC +GNC
-0.85
-2.71
169
19.15
Classic GNC
-1.12
-2.89
486
89.34
cuhe
IRIS GNC
-1.10
-2.90
141
26.13
Ours +GNC
-1.11
-2.66
86
7,86
Classic GNC
-0.80
-2.82
859
177.11
dragon
IRLS GNC
-0.80
-2.82
486
95.93
Ours' +GNC
-0.80
-2.80
201
26.42
Classic GNC
-0.88
-2.73
770
160.05
egyptian mask
IRLS GNC
-0.86
-2.75
264
53.51
Ours' +GNC
-0.87
-2.69
16.94
Classic GNC
-0.98
-2.87
713
148.55
sphere
IRLS GNC
-0.98
-2.88
220
45.73
Ours' +GNC
-0.99
-2.77
143
13.63
Classic GNC
-0.86
-2.84
765
159.25
vase
IRLS GNC
-0.87
-2.86
288
58.08
Ours' +GNC
-0.86
-2.77
17.05
The agent trained on the Aquarius sequence for point cloud regis-
tration task.
Table 9: Performance on GH non-convex function minimization benchmarks.
Problems
Method
f(x')
Iter
Time
Classic GHI
0.07
501
16.25
SLGH,
0.12
1839
56.71
SLGH,
0.26
28.45
2d Ackley
PGS
0.07
200
14.32
CPL
0.01
1701.61 61
2GH
0.05
359
12.31
Classic GH
0.00
501
11.39
SLGH,
0.00
1839
41.70
SLGH,
2.57
75
2.57
Himmelhlau
PGS
1.18
200
11.33
CPL
0.00
2160.17
Ours
0.00
8.91
Classic GH
0.00
501
23.76
SLGH,
0.00
1839
78.21
SLGH.,
0.34
319
19.64
Rastrigin
PGS
0.14
200
11.94
CPL
0.57
790.38
OGH
0.00
247
11.84
Classic GH
0.01
27.58
SLGH,
0.02
1839
91.90
10d Ackley
SLGH, 0.37
435
33.58
OGr 0.47
398
10.88
The agent trained on the Ackley functions with ran- domized parameters and evaluated on the canonical fixed- parameter version.
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Published as a conference paper at ICLR 2026
rotation_ene error (bunny)
(burny)
-0.5
Pgl0rrrattoon
-1.0
-1.5
50
GMC
Clsssiall GNC
GNC-IRLS
RL GNC
Clessical GNC
GNC-IRLS
(a) Rotation error of the bunny sequence in the point cloud registration task.
(b) Runtime of the bunny sequence in the point cloud registration task.
rotation emerr (cube)
Time (cube]
-1.0
100
Time
11.5
-2.0
GMC
Classical GNC
GNC-IRLS
AL GNC
Classical GNC
GNC-IRLS
(c) Rotation errоr of the sequence point cloud registration task.
(d) Runtime cube sequence the point cloud
registration task.
(reichstag)
(telchstag)
100
Time
50
RL GNC
Classical
GNG-IRLS
Classical GNC
GNC-IRLS
(e) Error of the reichstag sequence in the multi-view triangulation task.
(f) Runtime of the reichstag sequence in the multi-
view triangulation task.
error (sacre, coeur)
Time (sacre_ coeur)
Ing10le
-5
RL GNC
Classical GNC
GNC-IRLS
R1. GNC
Classical GNC
GNC-IRLS
(g) Error the sacre .coeur sequence the multi-view triangulation task.
Runtime of the sacre_coeur sequence in the multi- view triangulation task.
func (ackley)
Time lackley)
200
Time
100
0.0
RL
Clessic
SLGK
SLGH
PGS
RL
Clessic GH
S
SUGH d
PGS
(i) Function value of the Ackley problem in the non- convex function minimization task.
(j) Runtime of the Ackley problem in the non-convex function minimization task.
func (himmelblau)
Time (himmelblau)
40
lunc
20
Classic GH
S
SLGH
PCS
RL GH
Classic GH
SUGH
SLGH
PGS
(k) Function value the Himmelblau problem the ()) Runtime of the Himmelblau problem in the non- non-convex function minimization task.
convex function minimization task.
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 sta- bility and efficiency of each method.
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Published as a conference paper at ICLR 2026
Table 10: Performance on HC polynomial system benchmarks. Succ, denotes the success of tracking to root, and Time reports the average tracking time per solution path.
Problems
Method
Suce.
Iter
Time
Classic HC
100%
39
2.22
katsura10
OUC +HC
100%
0,65
Classic HC
100%
41
1.96
cyclic7
Ours++HC
100%
0.64
Classic HC
100%
41
1.69
noon5
OICC
100%
10
0.69
Classic HC Ours
100%
31
3.24
chandra9
100%
0.76
Classic HC
100%
53
8.25
UPnP
Simulator HC
100%
100
Ours*+II
100%
29
3.86
Runtimes not directly comparable, Simulator implemented in C++, while the other methods are in Python.
3 The agent is trained on separate set of polynomial systems with randomized coefficients.
F THE USE OF LARGE LANGUAGE MODELS (LLMs)
We use LLMs to polish writing.
22


Reviews:
[{"review_id": "9pvpZkuqWm", "reviewer": "Reviewer_7Gmc", "summary": "This paper proposes a Neural Predictor-Corrector (NPC) framework that leverages reinforcement learning (RL) to address homotopy problems across diverse domains, including robust optimization, global optimization, polynomial root-finding, and sampling. The core idea is to unify these traditionally independent homotopy tasks under a shared predictor-corrector (PC) structure, replacing hand-crafted heuristics for step-size selection and iteration termination with RL-learned adaptive policies. The authors employ an amortized training regime to enable one-time offline training and deployment on unseen instances, and validate NPC through experiments showing improved efficiency and stability compared to classical baselines.", "questions": "1. Generalization to Non-Homotopy PC Tasks: The paper emphasizes unification across homotopy problems, but many non-homotopy tasks (e.g., iterative convex optimization, SGD with adaptive learning rates) also use PC-like structures. Does NPC’s RL policy generalize to these non-homotopy PC tasks, or is it inherently tied to the homotopy interpolation paradigm? If not, what limits its generalizability?\n2. Comparison to Learning-Based PC Baselines: The paper compares NPC to classical PC methods but only briefly mentions learning-based baselines (e.g., Simulator HC for polynomial root-finding). Could the authors include a more detailed comparison to these learning-based alternatives?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Unified Framework for Diverse Homotopy Tasks: The paper identifies and formalizes the common PC structure underlying homotopy problems in optimization, root-finding, and sampling—an insight that helps consolidate fragmented research in these domains and highlights potential generalizability across tasks.\n2. Empirical Validation Across Domains: The authors conduct comprehensive experiments on four representative homotopy tasks (Graduated Non-Convexity, Gaussian Homotopy, Homotopy Continuation, Annealed Langevin Dynamics) and provide detailed ablation studies (e.g., RL state component analysis) to support the effectiveness of NPC in improving efficiency while preserving solution accuracy.\n3. Amortized Training for Practical Deployment: The amortized training design addresses a key limitation of task-specific learning methods by enabling deployment on unseen instances without per-task fine-tuning, which enhances the practical utility of the framework for real-world applications.", "weaknesses": "1. Limited Novelty in RL for Optimization/Sampling: The core premise of applying RL to improve optimization or sampling workflows is not new. As noted in the paper’s related work, prior studies (e.g., Li, 2019; Belder et al., 2023; Ye et al., 2025) have already explored RL for adaptive parameter tuning, optimizer design, and schedule prediction in similar problem spaces. The paper does not sufficiently distinguish NPC from these existing RL-driven optimization/sampling frameworks beyond its focus on homotopy-specific PC structures.\n2. Incremental Improvement Over Traditional Methods: NPC largely builds on the well-established PC algorithm for homotopy problems and only replaces heuristic step-size/termination rules with RL policies—this constitutes a relatively minor modification rather than a paradigm shift. The framework does not introduce new theoretical insights into homotopy methods or RL for sequential decision-making; instead, it refines existing components with incremental adjustments, limiting its contribution to methodological advancement.\n3. Dependence on Manual Reward Scaling: A critical practical limitation is the need for manual tuning of reward scales for each problem instance (detailed in Appendix A), which undermines the framework’s claim of being a “general solver.” This manual step not only increases the barrier to deployment but also contrasts with the goal of automating heuristic-driven decisions—an issue that the paper acknowledges but does not meaningfully address beyond proposing future work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "74lI9KDZf7", "reviewer": "Reviewer_pEeU", "summary": "The homotopy paradigm is a principle to solve optimization problems by transporting solutions to problems from a simple family to a target problem domain. The paper discusses application areas in which this principle is applied: annealing in optimization and sampling. The predictor corrector (PC) method is a family of methods that solve homotopy problems by predicting a new location of a solution and then correcting that prediction. Existing PC approaches typically rely on a an interpolation schedule that must be chosen in advance. While some domains offer natural choices, others do not as the authors claim. Therefore, the authors propose to learn an interpolation using RL, which is implemented by learning the predictor and corrector steps using a neural network. The evaluation of the paper contains tasks from 4 domains and ablation studies that showcase the generalizability of the approach and competitive performance. The ablations provide some insights into the importance of the individual components.", "questions": "- How do you weigh the two terms in your reward formulation? A specific equation would be very helpful.\n- For how long did you train your model? I am not familiar with the defaults of stable baselines and the info is not listed in the appendix.\n- How did you choose your kernel for computing the KSD in the evaluation?\n- As you state yourself, there are prior methods that use learning in the context of PC, but which seem to not generalize as well as yours as you point out in line 111. It would be interesting to see a comparison here, to understand better at which cost the generalization of your method comes, if an at all, especially as it is one of the key claimed contributions. Have you done such comparisons?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper provides a nice unifying perspective that was at least new to me.\n- The method seems to improve over baselines on the reported problems. For some of the domains they evaluate on (GNC, root finding) I am not sure how well chosen the baselines are but that is rather on me.\n- The method is pretty simple, and thus seems to be easy to reimplement. I am surprised that almost no hyperparameters must be changed from the stablebaselines defaults.\n- Overall the paper is pretty clear and well written.", "weaknesses": "### Method\n- The paper presents a very interesting and simple idea: use (reinforcement) learning to improve optimization and sampling methods. While I am not aware of any paper that discusses this under a single umbrella of homotopy problems, I have seen works that use learning to propose sampling steps, eg [1, 2, 3, 4, 5]. Some of these methods are mentioned in the related work section, yet I think it should become more clear that the objective of the paper is a unifying perspective.\n\n### Wording:\n- \"Because the predictor-corrector procedure is non-differentiable and early decisions influence the entire trajectory, supervised or self-supervised training is inadequate\" I think its totally fair that you use RL, but if you had data for supervised training, I think you could actually train in supervised fashion as your NN will be differentiable, no?\n- I am not sure if you can call the predictor and corrector schedules in Song et al. (2020) \"handcrafted heuristics \" as you state in your text, but their design choices are theoretically well motivated. Generally, using a linear schedule makes sense for many problems I would say.\n\n### Evaluation\n- The global optimzation problems are only in 2d, which seems pretty low to me, given that other communities like the derivative-free optimization community optimize on those functions in >10d. I would be interested in seeing the results on higher dimensions. But I think this point alone is not enough for rejection in my eyes.\n- I think the results are not the strongest on every task, for instance on the sampling problems, PGS seems to be on par at least. But I think this is a nitpick, since the approach seems to be more motivated to convince in its generality.\n\n### Clarity\n- While the high level idea of the paper is clear to me, some details of the algorithm are not: Why do you predict the corrector actions in line 3 of the algorithm? Is it always that single action you apply in line 7? If you predict both steps at once why have them separate at all? It would make much more sense to me if you first predicted the predictor step and then iteratively multiple different corrector steps based on H.\n\n### Minor\n- You should mention in the main paper that the functions you test on are 2d in Section 5.3.\n\n---\n### Sources\n[1] Richter, Lorenz, and Julius Berner. \"Improved sampling via learned diffusions.\" _arXiv preprint arXiv:2307.01198_ (2023).\n\n[2] Wang, Congye, et al. \"Reinforcement learning for adaptive MCMC.\" _arXiv preprint arXiv:2405.13574_ (2024).\n\n[3] Xi Lin, Zhiyuan Yang, Xiaoyuan Zhang, and Qingfu Zhang. Continuation path learning for homotopy optimization. In International Conference on Machine Learning, pp. 21288–21311. PMLR, 2023.\n\n[4] Ichikawa, Yuma. \"Controlling continuous relaxation for combinatorial optimization.\" _Advances in Neural Information Processing Systems_ 37 (2024): 47189-47216.\n\n[5] Hruby, Petr, et al. \"Learning to solve hard minimal problems.\" _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_. 2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lJfAjMiaho", "reviewer": "Reviewer_CRzY", "summary": "A number of problems in different domains can be framed as homotopy\ninterpolations between an easily solved source problem and a complex target\nproblem. Such problems are typically solved with a predictor-corrector\nframework. Here, a solution is found to the initial easy problem. Then, the\npredictor advances the problem following the homotopy interpolation, and the\ncorrector modifies the solution to fit the new problem.\n\nPrior work has treated these homotopy problems separately in different domains.\nThis paper proposes to unite a number of these problems under a single\nframework. It then proposes a neural predictor-corrector framework, which uses\nreinforcement learning to train a policy that outputs the next level for the\npredictor and the next tolerance for the corrector --- previously, these values\nwere determined via heuristics. Experiments show that the final RL agent\neffectively learns to set these values.", "questions": "1. I am unclear if the Homotopy Paradigm discussed in Sec. 3.1 is a novel\n   contribution of this work. Is the term \"Homotopy Paradigm\" already widely\n   used in the literature? If so, could a citation for it be added in Sec. 3.1?\n   I think I am confused because the first sentence of the introduction\n   (line 32) seems to indicate that the Homotopy paradigm is already well-known,\n   while lines 121-122 indicate that it is a perspective introduced in this\n   work. Assuming this perspective is novel, it may be good to list it as a\n   contribution in the introduction. Right now, the first contribution of the\n   introduction (line 82) makes it clear that unifying the approaches under the\n   homotopy paradigm is novel, but it is unclear if the homotopy paradigm itself\n   is novel.\n2. Based on Appendix D, it seems the NPC framework shifts the manual effort from\n   tuning the parameters for the predictor and corrector to tuning the reward\n   scales for the RL training. What do you see as the benefits and tradeoffs of\n   this shift? Might there be cases where it is easier to just use the\n   heuristics? When is it better to try to tune the rewards?\n3. Table 6 shows that removing certain components from the state can increase\n   the number of iterations required more than removing other components, e.g.,\n   removing the corrector's tolerance increases by 64, while removing the\n   homotopy level only increases by 21. Does this say anything about which parts\n   of the state are most essential to NPC?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This application of reinforcement learning seems quite novel to me. I\n   appreciate that PPO is able to be used \"out of the box\" to solve the problem.\n2. The problems chosen for the paper are quite diverse, which shows the\n   applicability of the framework. For example, I could see the application to\n   sampling / Langevin dynamics could eventually tie into generative modeling.\n3. The computational requirements for this method are quite low (lines 314-315),\n   making the research more accessible.\n4. The limitation of sensitivity to reward scale is acknowledged in the\n   appendix.\n5. The ablation of the state components helps in showing that all parts of the\n   state are necessary for NPC.\n\nOverall, I found the paper quite easy to read despite not having much\nfamiliarity with homotopy interpolations myself.", "weaknesses": "My main concern is that it seems the experiments were only conducted over one\ntrial, as I could not find any mention of repetitions, and no error bars are\nreported in the tables. It is advisable to conduct multiple trials of each\nalgorithm and use statistical testing to check that differences between\nalgorithms are significant.\n\nI have also listed several questions in the section below; these are minor points that I think would be useful to address in the updated paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)