On-policy distillation (OPD) trains a student on its own trajectories with token-level teacher feedback and often outperforms off-policy distillation and reinforcement learning with verifiable rewards. However, we find that its standard advantage-weighted policy gradient suffers from three structural weaknesses, including high-variance updates, vanishing gradients in zero-advantage regions, and exploration bottlenecks for models with limited capability. We therefore propose asymmetric on-policy distillation (AOPD), which replaces ineffective policy gradient with localized divergence minimization in non-positive advantage regions while preserving positive reinforcement learning. Experiments on mathematical reasoning benchmarks show that AOPD consistently outperforms standard OPD, with average gains of 4.09 / 8.34 under strong / weak initialization, respectively. AOPD also maintains higher policy entropy during training and better capability retention during sequential tool-use adaptation. Our source code is available at [repository](https://github.com/stargazer319/AOPD).

In this paper, we present Linguistics Informed Multi-Task BERT (LIMIT-BERT) for learning language representations across multiple linguistics tasks by Multi-Task Learning. LIMIT-BERT includes five key linguistics tasks: Part-Of-Speech (POS) tags, constituent and dependency syntactic parsing, span and dependency semantic role labeling (SRL). Different from recent Multi-Task Deep Neural Networks (MT-DNN), our LIMIT-BERT is fully linguistics motivated and thus is capable of adopting an improved masked training objective according to syntactic and semantic constituents. Besides, LIMIT-BERT takes a semi-supervised learning strategy to offer the same large amount of linguistics task data as that for the language model training. As a result, LIMIT-BERT not only improves linguistics tasks performance but also benefits from a regularization effect and linguistics information that leads to more general representations to help adapt to new tasks and domains. LIMIT-BERT outperforms the strong baseline Whole Word Masking BERT on both dependency and constituent syntactic/semantic parsing, GLUE benchmark, and SNLI task. Our practice on the proposed LIMIT-BERT also enables us to release a well pre-trained model for multi-purpose of natural language processing tasks once for all.