Approximation of Log-Partition Function in Policy Mirror Descent Induces Implicit Regularization for LLM Post-Training
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912881825546240 |
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| author | Xu, Zhenghao Lu, Qin Yu, Changlong Zhao, Tuo |
| author_facet | Xu, Zhenghao Lu, Qin Yu, Changlong Zhao, Tuo |
| contents | Policy mirror descent (PMD) provides a principled framework for reinforcement learning (RL) by iteratively solving KL-regularized policy improvement subproblems. While this approach has been adopted in training advanced LLMs such as Kimi K1.5/K2, the ideal closed-form PMD updates require reliable partition function estimation, a significant challenge when working with limited rollouts in the vast action spaces of LLMs. We investigate a practical algorithm, termed PMD-mean, that approximates the log-partition term with the mean reward under the sampling policy and performs regression in log-policy space. Specifically, we characterize the population solution of PMD-mean and demonstrate that it implicitly optimizes mirror descent subproblems with an adaptive mixed KL--$χ^2$ regularizer. This additional $χ^2$ regularization constrains large probability changes, producing more conservative updates when expected rewards are low and enhancing robustness against finite-sample estimation errors. Experiments on math reasoning tasks show that PMD-mean achieves superior performance with improved stability and time efficiency. These findings deepen our understanding of PMD-mean and illuminate pathways toward principled improvements in RL algorithms for LLMs. Code is available at https://github.com/horizon-rl/OpenKimi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05933 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Approximation of Log-Partition Function in Policy Mirror Descent Induces Implicit Regularization for LLM Post-Training Xu, Zhenghao Lu, Qin Yu, Changlong Zhao, Tuo Machine Learning Policy mirror descent (PMD) provides a principled framework for reinforcement learning (RL) by iteratively solving KL-regularized policy improvement subproblems. While this approach has been adopted in training advanced LLMs such as Kimi K1.5/K2, the ideal closed-form PMD updates require reliable partition function estimation, a significant challenge when working with limited rollouts in the vast action spaces of LLMs. We investigate a practical algorithm, termed PMD-mean, that approximates the log-partition term with the mean reward under the sampling policy and performs regression in log-policy space. Specifically, we characterize the population solution of PMD-mean and demonstrate that it implicitly optimizes mirror descent subproblems with an adaptive mixed KL--$χ^2$ regularizer. This additional $χ^2$ regularization constrains large probability changes, producing more conservative updates when expected rewards are low and enhancing robustness against finite-sample estimation errors. Experiments on math reasoning tasks show that PMD-mean achieves superior performance with improved stability and time efficiency. These findings deepen our understanding of PMD-mean and illuminate pathways toward principled improvements in RL algorithms for LLMs. Code is available at https://github.com/horizon-rl/OpenKimi. |
| title | Approximation of Log-Partition Function in Policy Mirror Descent Induces Implicit Regularization for LLM Post-Training |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.05933 |