Ordered 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.

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.

We propose a new perspective on policy optimization: rather than reweighting all samples by their importance ratios, an optimizer should select which samples are trustworthy enough to drive a policy update. Building on this view, we introduce Rejection-Gated Policy Optimization (RGPO), which replaces the importance sampling ratio r_theta = pi_theta / pi_old with a smooth, differentiable acceptance gate alpha_theta(s, a) = g(r_theta(s, a)) in the range [0, 1]. Unlike prior work that applies rejection sampling as a data-level heuristic before training, RGPO elevates rejection to an optimization principle: the gate participates directly in gradient computation and is implicitly updated alongside the policy. RGPO provides a unified framework: the policy gradients of TRPO, PPO, and REINFORCE all correspond to specific choices of the effective gradient weight w(r) = g'(r) * r. We prove that RGPO guarantees finite, bounded gradient variance even when importance sampling ratios are heavy-tailed (where IS variance diverges). We further show that RGPO incurs only a bounded, controllable bias and provides an approximate monotonic policy improvement guarantee analogous to TRPO. RGPO matches PPO in computational cost, requires no second-order optimization, and extends naturally to RLHF-style preference alignment. In online preference fine-tuning of Qwen2.5-1.5B-Instruct on Anthropic HH-RLHF (n = 3 seeds), RGPO uses a dual-ratio gate that anchors learning to both the previous policy and the reference model, achieving a Pareto-dominant outcome: the highest reward among online RL methods (+14.8% vs. PPO-RLHF) and the lowest KL divergence to the reference model (-16.0% vs. PPO-RLHF, -53.1% vs. GRPO).