A Distribution-Matching Perspective on LLM Training Algorithms

Reinforcement learning (RL) is widely used in language-model post-training to maximize rewards assigned to individual model outputs, such as scores from binary verifiers or reward models trained on human feedback. However, applications such as synthetic-data generation, fairness-related constraint satisfaction, and policy exploration require controlling the distribution of outputs across model generations rather than only maximizing expected reward. We propose a general RL-based framework for Distribution Matching allowing matching the distribution of a latent categorical attribute of model outputs to a specified target distribution. Empirically, we demonstrate that dominant post-training recipes such as Group Relative Policy Optimization (GRPO) reduce output diversity by concentrating policy probability towards a single mode. Entropy regularization and sampling temperature can improve the spread of the distribution but have constrained effectiveness, limited to apply only in token space and toward uniform distributions. We show that prior work in this area is a specific case of Distribution Matching involving the $L_2$ divergence. We then propose reward functions for other divergences such as KL and Jensen-Shannon and motivate them with theoretical justification. Finally, we demonstrate the effectiveness of our approach on a set of experiments involving mathematical reasoning and programming.

Since the advent of knowledge distillation, KL divergence has been the standard loss in distillation. Recently, on-policy distillation (OPD) has emerged as an efficient post-training paradigm for LLMs. As a distillation method, OPD naturally inherits KL divergence as its standard loss. However, in this work, we find that KL divergence may not be necessary for OPD. We show that simply preserving the update direction is sufficient for effective OPD. As long as the update direction is toward the teacher, OPD works. More precisely, it is not the direction of every token, but the direction of a small subset of tokens where the teacher and student disagree strongly. We first show that simply assigning a reward of (+1) to tokens where the teacher probability is higher than the student probability and (-1) where it is lower, which merely encourages updates toward the teacher, reproduces almost the same training mode as OPD with reverse KL. We further show that only the direction of a small subset of tokens with large teacher-student disagreement is critical, and training works as long as their update direction is toward the teacher, even if other tokens are pulled away from the teacher. And as an application of these findings, we introduce Consensus Multi-Teacher On-Policy Distillation (C-MOPD) to improve Multi-Teacher On-Policy Distillation (MOPD). Unlike MOPD, which routes each sample to a single teacher and may cause capability conflicts across domains, C-MOPD lets every sample be supervised by all teachers. Experiments show that C-MOPD consistently outperforms MOPD on both math and code benchmarks. Our code is available at https://github.com/LeapLabTHU/KL-Free-OPD.

On-policy distillation (OPD) is becoming an important component of large language model (LLM) post-training for transferring the reasoning capability of a strong teacher LLM to a weaker student LLM. OPD trains the student by minimizing the reverse KL divergence between the teacher and the student via rollouts generated by the student's policy. However, estimating the gradient of the reverse KL divergence in OPD remains a challenge. Using only the sampled token from the student-generated rollout is computationally cheap but provides limited distributional supervision, which will degrade accuracy. In addition, using the full vocabulary provides complete distributional supervision but is computationally expensive. Therefore, recent works propose Top-$k$ OPD (TK-OPD) that use selected top-$k$ tokens, which provides richer distributional supervision than sampled-token estimation at substantially lower computational cost than full-vocabulary estimation. Unfortunately, using only the selected top-$k$ tokens induces bias, leading to accuracy degradation, as the probability mass outside the selected top-$k$ tokens is discarded. To address the bias of TK-OPD, we propose Tail-Corrected Top-$k$ On-Policy Distillation (TT-OPD). It preserves the advantages of TK-OPD, including rich distributional supervision and low computational cost, while providing an unbiased estimator of the gradient of the reverse KL divergence. The key insight of TT-OPD is to use not only the selected top-$k$ tokens, but also the sampled token from the student-generated rollout, thereby recovering the discarded probability mass in expectation, avoiding the bias. Experimental results demonstrate that TT-OPD significantly outperforms other tested OPD variants.