A Closed-Loop Non-Asymptotic Convergence Analysis of PPO with Learned Critics and Clipping

Proximal Policy Optimization (PPO) is the standard policy-gradient algorithm for on-policy reinforcement learning. The literature presents it in two forms, a clipped surrogate that bounds the importance ratio between successive policies and a Kullback-Leibler penalty between them. These forms are treated as separate algorithms with their own gradients, their own hyperparameters, and their own reference implementations, and a sizeable body of empirical work compares them. We show that the gradient of the clipped surrogate is reproduced exactly by a Kullback-Leibler surrogate whose coefficient varies per sample, with closed-form dependence on the importance ratio and the advantage. The identity holds at every minibatch step and across the entire inner loop, and on five MuJoCo continuous-control benchmarks the two losses produce indistinguishable training curves. The reformulation exposes a structural feature of the clipped surrogate that the min notation hides. PPO-Clip's implicit per-sample penalty is a step function at the boundary of the trust region, and the shape of this coefficient is the natural design axis for generalising the algorithm. We sketch the resulting follow-up directions in the discussion.

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.

Group relative policy optimization for reinforcement learning with verifiable rewards (RLVR) typically uses a fixed importance-sampling (IS) ratio clipping boundary across all rollouts. We identify a key limitation: rare correct rollouts on harder problems and abundant correct rollouts on easier problems are clipped at comparable rates, despite contributing very different learning signals. Rollouts with low group success exhibit larger IS ratios and carry stronger gradient signal for exploration and solving new problems, yet are disproportionately suppressed by fixed clipping. To address this, we propose Group Adaptive Clipping Policy Optimization (GAPO), a plug-in modification to GRPO methods that adapts the clipping boundary to the rollout advantage. GAPO is motivated by a reverse-KL trust-region perspective, which suggests that rollouts with larger learning signal should receive proportionally greater update headroom. GAPO requires no reward shaping and preserves the standard PPO/GSPO surrogate while adapting only the clipping threshold. Across Qwen and Llama models, GAPO consistently improves both Pass@1 and Pass@k over fixed clipping and advantage-shaping baselines on math reasoning and coding benchmarks where the pass rates by the base model are relatively low.