Understanding and Exploiting Anisotropy in Post-Training

We study how post-training changes the weights of Large Language Models (LLMs) relative to their pretrained weights. Across 12 post-training chains with supervised fine-tuning (SFT) and reinforcement learning (RL), we express each weight update in the pretrained matrix's singular value decomposition (SVD) frame. This decomposition separates the changes of three geometrically distinct components: diagonal values, which reshapes singular values; off-diagonal values, which rotates the coupling between pretrained input and output directions; and null-space values, which routes outside the matrix's original nonzero SVD core. On a math evaluation suite, we find that removing the diagonal component usually preserves most of the gains from post-training. These results suggest that post-training gains are carried primarily by reconfiguring and extending pretrained pathways rather than by substantially changing singular values of pre-trained models.

Reinforcement learning (RL)-based post-training has become an effective approach for eliciting reasoning capabilities in large language models (LLMs). However, adapting post-RL models to new knowledge domains or behaviors through subsequent supervised fine-tuning (SFT) can severely overwrite these capabilities. Existing approaches mitigate such forgetting through experience replay, specialized initialization, or constrained optimization using gradient projection, but either provide limited preservation or incur substantial training overhead. Our analysis shows that reasoning activations concentrate in low-dimensional subspaces, leaving substantial null-space capacity for adaptation, and that the corresponding approximate null spaces can be reliably estimated from a modest number of examples. Motivated by these observations, we propose Null-Basis Low-Rank Adaptation (NB-LoRA), a parameter-efficient method for adapting post-RL LLMs while preserving their acquired reasoning ability. We formulate reasoning retention as a layer-wise hidden-state preservation constraint and construct a fixed approximate null basis from reasoning activations. LoRA updates are then reparameterized through this basis, enforcing the preservation constraint throughout fine-tuning. Extensive experiments across multiple RL-trained LLMs and diverse downstream tasks show that NB-LoRA matches standard LoRA in adaptation performance, maintains reasoning accuracy near pre-fine-tuning levels, and generalizes this preservation to held-out reasoning benchmarks.

Reinforcement learning (RL) post-training has emerged as a powerful framework for enhancing the capabilities of large language models (LLMs), enabling impressive reasoning, math, and coding capabilities. Yet for many researchers and practitioners, the principles behind classical RL remain a "black box". In this work, we deconstruct the RL post-training algorithm, investigating each step to clarify what is actually happening beneath the surface. By isolating the mechanics of RL with Verifiable Rewards in a controlled and simplified environment, we examine how RL outcomes are shaped by the base model's prior distribution, the granularity of the reward signal, the diversity of the prompt distribution, and model scale. We use the entropy of the policy's output distribution as a lens to compare the distributions learned through pretraining, SFT, and RL post-training, revealing how each stage shapes model certainty. Our investigation sheds light on how these choices interact to affect post-training success. For example, we show that the effect of so-called 'spurious rewards' depends on the prompt distribution used for post-training. We also provide insight into why the success of RL post-training depends on whether the base model already places sufficient probability mass on the desired behavior, linking it to the classical concept of exploration in RL. Ultimately, we provide this primer as a resource to those in the NLP community wishing to incorporate RL as a tool in their toolbox.