Implicit Actor Critic Coupling via a Supervised Learning Framework for
  RLVR
Authors: name: Jiaming Li, name: Longze Chen, name: Ze Gong, name: Yukun Chen, name: Lu Wang, name: Wanwei He, name: Run Luo, name: Min Yang

Recent advances in Reinforcement Learning with Verifiable Rewards (RLVR) have
empowered large language models (LLMs) to tackle challenging reasoning tasks
such as mathematics and programming. RLVR leverages verifiable outcome rewards
to guide policy optimization, enabling LLMs to progressively improve output
quality in a grounded and reliable manner. Despite its promise, the RLVR
paradigm poses significant challenges, as existing methods often suffer from
sparse reward signals and unstable policy gradient updates, particularly in
RL-based approaches. To address the challenges, we propose PACS, a
novel RLVR framework that achieves imPlicit Actor
Critic coupling via a Supervised learning framework. By
treating the outcome reward as a predictable label, we reformulate the RLVR
problem into a supervised learning task over a score function parameterized by
the policy model and optimized using cross-entropy loss. A detailed gradient
analysis shows that this supervised formulation inherently recovers the
classical policy gradient update while implicitly coupling actor and critic
roles, yielding more stable and efficient training. Benchmarking on challenging
mathematical reasoning tasks, PACS outperforms strong RLVR baselines, such as
PPO and GRPO, achieving superior reasoning performance. For instance, PACS
achieves 59.78\% at pass@256 on AIME 2025, representing improvements of 13.32
and 14.36 points over PPO and GRPO. This simple yet powerful framework offers a
promising avenue for LLMs post-training with verifiable rewards. Our code and
data are available as open source at https://github.com/ritzz-ai/PACS.