ANO: Robust Policy Optimization via Bounded, Redescending Gain Fields

Yiheng Zhang; Yiming Wang; Kaiyan Zhao; Zhenglin Wan; Jiayu Chen; Leong Hou U

cs.AIMay 4, 2026

ANO: Robust Policy Optimization via Bounded, Redescending Gain Fields

Authors: Yiheng Zhang, Yiming Wang, Kaiyan Zhao, Zhenglin Wan, Jiayu Chen, Leong Hou U

Organizations: University of Macau · Wuhan University · National University of Singapore · University of Hong Kong

Abstract

Proximal Policy Optimization (PPO) dominates reinforcement learning and LLM alignment, yet its hard-clipping mechanism and unconstrained alternatives (e.g., SPO) sit at two extremes of a stability-efficiency dilemma. We argue that this dilemma is best understood dynamically: a surrogate objective is a feedback law on the probability ratio, and its clipping/penalty shape defines a gain field that drives the update dynamics. PPO's clip induces a dead zone (zero feedback outside the trust region), leaving the policy to drift open-loop under momentum; SPO's quadratic penalty induces an unbounded, linearly growing gain that stiffens the dynamics and destabilizes under aggressive step sizes. Guided by this view, we derive Anchored Neighborhood Optimization (ANO), which designs the gain field directly: a $C^\infty$ shaping kernel that anchors the identity map at $r{=}1$ , peaks exactly at a prescribed trust-region boundary $1{+}ε$ , bounds the push on severely off-policy samples by a tunable $κ_{+}$ , and exerts a bounded, redescending pull of tunable depth $κ_{-}$ on extreme outliers. The three hyperparameters have decoupled roles, and all internal constants are solved in closed form. Empirically, ANO ranks first on both Atari (40 games) and MuJoCo in IQM and Median of normalized scores. While the runner-up differs across domains (PAPO on Atari, SPO on MuJoCo), ANO is the only method consistently at the top. Under a learning-rate stress test ( $3\times10^{-4}\!\to\!10^{-3}$ ), ANO degrades by only $0.9\%$ whereas PPO collapses by $54.5\%$ , and the stressed ANO still outperforms PPO and PAPO at their best-tuned learning rates.

Explore similar work

Jul 11, 2026cs.LG

Beyond Euclidean Clipping: Overcoming Exploration Collapse in LLM RL via Riemannian Isometric Policy Optimization

Reinforcement learning (RL) has become a dominant paradigm for enhancing LLMs' reasoning capabilities. However, RL algorithms with PPO-Clip are inherently limited by exploration collapse. Subsequent works remain primarily heuristic and fail to identify the essential cause of PPO-Clip's failure. This work reveals the fundamental flaw of PPO-Clip: it implicitly measures policy discrepancy using Euclidean metric, which is theoretically inconsistent with the intrinsic geometry on the policy Riemannian manifold. This geometric mismatch results in overly conservative updates in low-probability regions while aggressive in high-probability regions, ultimately collapsing exploration. To correct this geometric flaw, we propose Riemannian Isometric Policy Optimization (RIPO), which guarantees isometric policy updates on the Riemannian manifold, effectively balancing exploration and exploitation. We further show that RIPO achieves a favorable bias-variance trade-off, which stabilizes optimization. Extensive experiments demonstrate that RIPO significantly surpasses existing LLM RL algorithms across seven competition-level benchmarks (up to 60% improvement over GRPO on AIME24).

Zhicheng Cai, Xinyuan Guo, Hanlin Wu +4

Gradient Clipping Riemannian Optimization

Ask PDF

Apr 20, 2026cs.LG

Bounded Ratio Reinforcement Learning

Proximal Policy Optimization (PPO) has become the predominant algorithm for on-policy reinforcement learning due to its scalability and empirical robustness across domains. However, there is a significant disconnect between the underlying foundations of trust region methods and the heuristic clipped objective used in PPO. In this paper, we bridge this gap by introducing the Bounded Ratio Reinforcement Learning (BRRL) framework. We formulate a novel regularized and constrained policy optimization problem and derive its analytical optimal solution. We prove that this solution ensures monotonic performance improvement. To handle parameterized policy classes, we develop a policy optimization algorithm called Bounded Policy Optimization (BPO) that minimizes an advantage-weighted divergence between the policy and the analytic optimal solution from BRRL. We further establish a lower bound on the expected performance of the resulting policy in terms of the BPO loss function. Notably, our framework also provides a new theoretical lens to interpret the success of the PPO loss, and connects trust region policy optimization and the Cross-Entropy Method (CEM). We additionally extend BPO to Group-relative BPO (GBPO) for LLM fine-tuning. Empirical evaluations of BPO across MuJoCo, Atari, and complex IsaacLab environments (e.g., Humanoid locomotion), and of GBPO for LLM fine-tuning tasks, demonstrate that BPO and GBPO generally match or outperform PPO and GRPO in stability and final performance.

Yunke Ao, Le Chen, Bruce D. Lee +5

Proximal Policy Optimization Deep Reinforcement Learning Trust Region

Ask PDF

May 26, 2026cs.LG

Ratio-Variance Regularized Policy Optimization

Standard on-policy reinforcement learning relies on heuristic clipping to enforce trust regions, but this mechanism imposes a severe cost by indiscriminately truncating high-return yet high-divergence updates. We demonstrate that explicitly constraining the policy ratio variance provides a principled local approximation to trust-region constraints, eliminating the need for binary hard clipping. By acting as a distributional ``soft brake'', this approach preserves critical gradient signals from novel discoveries while naturally down-weighting and enabling the reuse of stale, off-policy data. We introduce

{\bf R}^2{\bf VPO}

(Ratio-Variance Regularized Policy Optimization), which implements this constraint via a primal-dual optimization framework. Extensive evaluations across

7

LLM scales, spanning both fast and slow reasoning paradigms, and

10

robotic control tasks demonstrate the generality of the proposed approach. R

^2

VPO achieves substantial performance gains on mathematical reasoning benchmarks, with particularly pronounced improvements on smaller models, while significantly improving sample efficiency. Furthermore, it consistently outperforms PPO baselines in continuous control domains, particularly in sparse-reward and dynamic environments. Together, these findings establish ratio-variance regularization as a principled foundation for stable and data-efficient policy optimization.

Yu Luo, Shuo Han, Yihan Hu +5

Trust Region

Ask PDF