Self-Improvement with Auxiliary Rewards: Learning from Verified Traces and Evaluating Transfer

When a Large Language Model (LLM) is trained with Reinforcement Learning (RL) on multiple tasks, a task's primary reward can gate auxiliary rewards such as a length reward: on a failing response, an auxiliary cannot exceed the primary. However, group-relative advantage estimators treat a gated auxiliary like any other component, so it can reverse the primary's order, and at the primary's ceiling only auxiliaries order responses. Keeping gated auxiliaries hurts some tasks, while dropping them hurts others. We propose CONCUR, an advantage estimator that replaces the standardized scores of each gated auxiliary with the closest scores that satisfy two conditions: responses with higher primary rewards cannot receive lower scores, and responses at the primary's ceiling receive equal scores. CONCUR-T applies both conditions throughout training, and CONCUR-A lifts the second for an auxiliary that agrees with the primary's order more often than it reverses it among passing responses. On three models from two families and across math, code, instruction following, and writing, both variants score higher on average than all baselines and than when all the gated auxiliaries are dropped.

Can a language model improve reasoning by learning from its own imperfect responses, without rewards or teacher-provided solutions? We present Self-evolving Post-Training (SePT), a simple method that alternates temperature-controlled self-generation with next-token likelihood training. Each round uses the updated model to generate new training responses, with one response per prompt by default and no correctness filtering. Across six mathematical benchmarks, SePT improves a temperature-selected no-training baseline by 11.4 and 6.7 AVG points on Qwen2.5-Math-7B and Qwen2.5-7B, respectively, where AVG averages Pass@1, Pass@8 and Pass@32 across benchmarks. We analyze how sampling temperature shapes the learning signal and investigate the value of the resulting responses. Responses from a SePT-trained model improve a student initialized from the original weights, while their reasoning prefixes help an unchanged model complete solutions, even when matched in length to prefixes from a colder initial model. Comparing next-token predictions at identical contexts also reveals changes in token rankings that decoding-temperature adjustment cannot reproduce. Further evaluations across nine starting models, general reasoning and code generation examine broader applicability. Together, these results show that reward-free self-training can improve both a model's predictions and the supervision it provides. Our code is available at https://github.com/ElementQi/SePT.

Reinforcement learning with verifiable rewards (RLVR) provides reliable outcome supervision for language model reasoning, but scalar trajectory rewards offer limited guidance for token-level credit assignment. Existing self-distillation methods incorporate privileged teacher information through fixed strategies: direct distribution matching may destabilize successful behavior, while magnitude modulation provides limited corrective guidance after failure. We propose Hybrid Hindsight Self-Distillation (H$^2$SD), which adapts both teacher context and update strategy to trajectory correctness. For successful trajectories, the teacher conditions on the verified student response and a rephrasing instruction to rescore the original tokens. These scores modulate reward-derived updates while preserving advantage signs. For failed trajectories, a verified reference hint guides corrective supervision through reverse KL distillation. Across Sudoku, Calcudoku, Arrow Maze, and mathematical reasoning with two Qwen3 models, H$^2$SD achieves an overall average accuracy of 43.33%, outperforming the strongest baseline by 11.43 percentage points while maintaining stable optimization.