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Policy Newton Algorithm in Reproducing Kernel Hilbert Space

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.7Download Report PDF
Reinforcement learningRKHSNewton method
Abstract:

Reinforcement learning (RL) policies represented in Reproducing Kernel Hilbert Spaces (RKHS) offer powerful representational capabilities. While second-order optimization methods like Newton's method demonstrate faster convergence than first-order approaches, current RKHS-based policy optimization remains constrained to first-order techniques. This limitation stems primarily from the intractability of explicitly computing and inverting the infinite-dimensional Hessian operator in RKHS. We introduce Policy Newton in RKHS, the first second-order optimization framework specifically designed for RL policies represented in RKHS. Our approach circumvents direct computation of the inverse Hessian operator by optimizing a cubic regularized auxiliary objective function. Crucially, we leverage the Representer Theorem to transform this infinite-dimensional optimization into an equivalent, computationally tractable finite-dimensional problem whose dimensionality scales with the trajectory data volume. We establish theoretical guarantees proving convergence to a local optimum with a local quadratic convergence rate. Empirical evaluations on a toy financial asset allocation problem validate these theoretical properties, while experiments on standard RL benchmarks demonstrate that Policy Newton in RKHS achieves superior convergence speed and higher episodic rewards compared to established first-order RKHS approaches and parametric second-order methods. Our work bridges a critical gap between non-parametric policy representations and second-order optimization methods in reinforcement learning.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper introduces a second-order optimization framework for reinforcement learning policies represented in reproducing kernel Hilbert spaces, specifically addressing the challenge of computing and inverting infinite-dimensional Hessian operators. According to the taxonomy, this work resides in the 'Policy Optimization with Second-Order RKHS Methods' leaf, which currently contains only this paper as its sole member. This positioning suggests the paper occupies a relatively sparse research direction within the broader field of RKHS-based policy optimization, where most existing work employs first-order or evolutionary approaches.

The taxonomy reveals that neighboring research directions include variational inference with second-order RKHS methods and positive function optimization using pseudo-mirror descent, both of which leverage curvature information but target different problem classes. The paper's approach diverges from the more populated 'First-Order and Evolutionary Policy Optimization in RKHS' branch, which encompasses gradient-based proximal methods and covariance matrix adaptation strategies. The taxonomy structure indicates that while second-order methods exist for related tasks like Stein variational inference, direct application to policy optimization in RKHS remains underexplored, with the paper attempting to bridge this gap.

Among the three identified contributions, the literature search examined 24 candidates total. The core algorithmic contribution (cubic regularization with finite-dimensional reduction) was assessed against 10 candidates, with 1 appearing to provide overlapping prior work. The theoretical convergence guarantees were evaluated against 10 candidates, with 2 potentially offering similar results. Notably, the claim of being the 'first second-order optimization framework for RKHS policies' was examined against 4 candidates with no clear refutations found. These statistics reflect a limited search scope rather than exhaustive coverage, suggesting that while some technical components have precedent, the specific integration for policy optimization may represent a novel synthesis.

Based on the available signals from 24 examined candidates, the work appears to occupy a genuinely sparse area within the taxonomy, though the limited search scope prevents definitive conclusions about absolute novelty. The absence of sibling papers in its taxonomy leaf and the mixed refutation results across contributions suggest the paper combines known techniques in a potentially novel configuration, though comprehensive assessment would require broader literature coverage beyond the top-K semantic matches analyzed here.

Taxonomy

Core-task Taxonomy Papers
6
3
Claimed Contributions
24
Contribution Candidate Papers Compared
3
Refutable Paper

Research Landscape Overview

Core task: second-order policy optimization in reproducing kernel Hilbert space. The field structure reflects a spectrum of approaches for learning policies in RKHS, ranging from computationally intensive second-order methods that exploit curvature information to more scalable first-order and evolutionary strategies. The taxonomy organizes work into several main branches: one focuses on second-order optimization methods that leverage Hessian or Fisher information within the RKHS framework, another addresses online kernel learning with mechanisms to manage growing model complexity, a third encompasses first-order gradient-based and derivative-free evolutionary techniques, and a fourth examines integral reinforcement learning with an eye toward computational tractability. Representative works such as Stein Variational Newton[3] and Second-Order Online Kernel[5] illustrate how curvature-aware updates can be formulated in infinite-dimensional spaces, while methods like CMA-ES Direct Policy RKHS[1] demonstrate evolutionary alternatives that sidestep explicit gradient computation. A particularly active line of inquiry concerns the trade-off between sample efficiency and computational overhead: second-order methods promise faster convergence by incorporating curvature, yet they often require expensive matrix operations or approximations to remain feasible in high-dimensional or online settings. Policy Newton RKHS[0] sits squarely within the branch of second-order RKHS methods, emphasizing Newton-type updates that exploit the geometry of the policy space. Its approach contrasts with lighter-weight schemes such as Sparse Pseudo-Mirror Descent[4], which sacrifices some curvature information to maintain sparsity and scalability, and with Computation Integral Reinforcement Learning[2], which prioritizes computational considerations in a related but distinct integral formulation. By adopting a full second-order perspective, Policy Newton RKHS[0] aligns closely with works that seek principled curvature exploitation, positioning itself as a rigorous yet computationally demanding option among the spectrum of RKHS-based policy optimization techniques.

Claimed Contributions

Policy Newton in RKHS algorithm with cubic regularization and finite-dimensional reduction

The authors introduce the first second-order optimization method for RL policies in RKHS by deriving the Hessian operator as a second-order Fréchet derivative and using a cubic regularized auxiliary function to avoid computing the intractable inverse. They leverage the Representer Theorem to transform the infinite-dimensional optimization into a tractable finite-dimensional problem whose dimension scales with trajectory data volume.

10 retrieved papers
Can Refute
Theoretical convergence guarantees with local quadratic convergence rate

The authors provide theoretical analysis proving that Policy Newton in RKHS converges to a local optimum and achieves a local quadratic convergence rate, establishing formal guarantees for the second-order method in the RKHS setting.

10 retrieved papers
Can Refute
First second-order optimization framework for RKHS policies in reinforcement learning

The work bridges a critical gap by developing the first second-order optimization framework tailored for reinforcement learning policies represented in Reproducing Kernel Hilbert Spaces, addressing the limitation that previous RKHS policy optimization was constrained to first-order methods.

4 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Policy Newton in RKHS algorithm with cubic regularization and finite-dimensional reduction

The authors introduce the first second-order optimization method for RL policies in RKHS by deriving the Hessian operator as a second-order Fréchet derivative and using a cubic regularized auxiliary function to avoid computing the intractable inverse. They leverage the Representer Theorem to transform the infinite-dimensional optimization into a tractable finite-dimensional problem whose dimension scales with trajectory data volume.

Contribution

Theoretical convergence guarantees with local quadratic convergence rate

The authors provide theoretical analysis proving that Policy Newton in RKHS converges to a local optimum and achieves a local quadratic convergence rate, establishing formal guarantees for the second-order method in the RKHS setting.

Contribution

First second-order optimization framework for RKHS policies in reinforcement learning

The work bridges a critical gap by developing the first second-order optimization framework tailored for reinforcement learning policies represented in Reproducing Kernel Hilbert Spaces, addressing the limitation that previous RKHS policy optimization was constrained to first-order methods.

Table of Contents

Policy Newton Algorithm in Reproducing Kernel Hilbert Space | Novelty Validation