Post-Training with Policy Gradients: Optimality and the Base Model Barrier
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908871479525376 |
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| author | Mousavi-Hosseini, Alireza Erdogdu, Murat A. |
| author_facet | Mousavi-Hosseini, Alireza Erdogdu, Murat A. |
| contents | We study post-training linear autoregressive models with outcome and process rewards. Given a context $\boldsymbol{x}$, the model must predict the response $\boldsymbol{y} \in Y^N$, a sequence of length $N$ that satisfies a $γ$ margin condition, an extension of the standard separability to sequences. We prove that on test samples where the base model achieves a non-trivial likelihood $α$, a variant of policy gradient (PG) can achieve likelihood $1 - \varepsilon$ with an essentially minimax optimal number of reward queries $\tilde{O}((α^{-1} + \varepsilon^{-1})/γ^2)$. However, a barrier arises for going beyond the support of the base model. We prove that the overall expected error after post-training with outcome rewards is governed by a property of the base model called the Likelihood Quantile (LQ), and that variants of PG, while minimax optimal, may require a number of reward queries exponential in $N$ to go beyond this support, regardless of the pre-training algorithm. To overcome this barrier, we study post-training with a process reward model, and demonstrate how PG variants in this setting avoid the curse of dimensionality in $N$ via dependence on a token-level LQ. Along the way, we prove that under the margin condition, SGD with adaptive learning rate (LR) achieves a near optimal test error for statistical learning, and PG with adaptive LR achieves a near optimal number of mistakes for online learning while being computationally efficient whenever possible, both of which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_06957 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Post-Training with Policy Gradients: Optimality and the Base Model Barrier Mousavi-Hosseini, Alireza Erdogdu, Murat A. Machine Learning Artificial Intelligence We study post-training linear autoregressive models with outcome and process rewards. Given a context $\boldsymbol{x}$, the model must predict the response $\boldsymbol{y} \in Y^N$, a sequence of length $N$ that satisfies a $γ$ margin condition, an extension of the standard separability to sequences. We prove that on test samples where the base model achieves a non-trivial likelihood $α$, a variant of policy gradient (PG) can achieve likelihood $1 - \varepsilon$ with an essentially minimax optimal number of reward queries $\tilde{O}((α^{-1} + \varepsilon^{-1})/γ^2)$. However, a barrier arises for going beyond the support of the base model. We prove that the overall expected error after post-training with outcome rewards is governed by a property of the base model called the Likelihood Quantile (LQ), and that variants of PG, while minimax optimal, may require a number of reward queries exponential in $N$ to go beyond this support, regardless of the pre-training algorithm. To overcome this barrier, we study post-training with a process reward model, and demonstrate how PG variants in this setting avoid the curse of dimensionality in $N$ via dependence on a token-level LQ. Along the way, we prove that under the margin condition, SGD with adaptive learning rate (LR) achieves a near optimal test error for statistical learning, and PG with adaptive LR achieves a near optimal number of mistakes for online learning while being computationally efficient whenever possible, both of which may be of independent interest. |
| title | Post-Training with Policy Gradients: Optimality and the Base Model Barrier |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2603.06957 |