Stabilizing Policy Optimization via Logits Convexity
Authors: Hongzhan Chen, Tao Yang, Yuhua Zhu, Shiping Gao, Xiaojun Quan, Ting Yao
Organizations: School of Computer Science and Engineering, Sun Yat-sen University, China · Shanghai Innovation Institute · Wechat Search, Tencent Inc, China · Shenzhen Loop Area Institute
While reinforcement learning (RL) has been central to the recent success of large language models (LLMs), RL optimization is notoriously unstable, especially when compared to supervised fine-tuning (SFT). In this work, we investigate the stability gap between SFT and RL from a gradient-based perspective, and show that the convexity of the SFT loss with respect to model logits plays a key role in enabling stable training. Our theoretical analysis demonstrates that this property induces favorable gradient directionality during optimization. In contrast, Proximal Policy Optimization (PPO), a widely adopted policy gradient algorithm utilizing a clipped surrogate objective, lacks this stabilizing property. Motivated by this observation, we propose Logits Convex Optimization (LCO), a simple yet effective policy optimization framework that aligns the learned policy with an optimal target derived from the original RL objective, thereby emulating the stabilizing effects of logits-level convexity. Extensive experiments across multiple model families show that our LCO framework consistently improves training stability and outperforms conventional RL methods on a broad range of benchmarks.
Figures & tables
Figure 1: (a) Gradient norm during training for SFT and PPO. (b) Pass@1 results of SFT and PPO on the MATH500 benchmark..
Figure 2: (a) Gradient norm during training for LCO-KLD. (b) Pass@1 results of LCO-KLD on MATH500.
Figure 3: Training dynamics of SFT. (a) Gradient norm decreases as training progresses while target token probability on training sample increases. The magnitudes of both target token logit gradient (b) and the non-target token logit gradient (c) diminish as training progresses.
Figure 4: Training dynamics of PPO for actions with negative advantages. (a) Gradient norm increases before being clipped, while sampled action probabilities decrease. The magnitudes of both sampled action logit gradient (b) and non-sampled action logit gradient (c) increase before being clipped. The results show a correlation between the behavior of parameter-space gradients and logit-space gradients.
Figure 5: Training dynamics of LCO-KLD for actions with negative advantages. Unlike training dynamics of PPO, LCO‑KLD exhibits stable gradient updates that scale with the remaining error and diminish progressively as the model converges.
Figure 6: Training dynamics of LCO-KLD. (a) Gradient norm for actions with positive advantages. (b) Gradient norm for actions with negative advantages. Unlike training dynamics of PPO, these results demonstrate a stable gradient update of LCO-KLD during training.
Figure 7: Training efficiency of LCO-LCH, LCO-KLD, and PPO, evaluated by policy accuracy on the MATH500 evaluation set.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 8: Training dynamics of PPO for actions with positive advantages. (a) Gradient norm decreases as training progresses while sampled action probabilities increase. The magnitudes of both sampled (b) and non-sampled action logit gradient (c) decrease.
Figure 9: Training dynamics of LCO-KLD for actions with positive advantages. We show the dynamics of gradient norm (a) , logit gradient for sampled (b) and non-sampled (c) actions. LCO‑KLD exhibits stable gradient updates as the model converges.
Shenzhen International Graduate School, Tsinghua University · Intelligent System Department, Zhongxing Telecom Equipment (ZTE) · Institute of Advanced Interdisciplinary Studies, Chongqing University of Posts and Telecommunications