GRRR: The Geometry of Reshaping, Rotation, and Routing in Decoder LLM post-training

LLM post-training combines supervised fine-tuning (SFT), a mode-covering forward-KL objective, with reinforcement learning (RL), a mode-seeking reverse-KL objective. Frequency-weighted likelihood training leaves a well-known signature: anisotropy, in which a few residual channels carry disproportionately large activations. Anisotropy is widely documented and usually treated as a defect, yet its function and its interaction with post-training remain unclear. We first analyze it. A label-free outlier rule isolates about 5% of residual channels that are essential for language modeling: removing them raises perplexity from 10 to over $10^6$, versus 35 for count-matched random channels. Yet they barely distinguish correct from incorrect reasoning. SFT reshapes them, whereas RL leaves them largely intact and adapts the complementary channels. These channels therefore form the model's coherence substrate, and reasoning adaptation happens elsewhere. We then exploit this. SphereGate learns one bounded gain per residual channel on a frozen backbone. Its activation-weighted gradients provably limit movement of high-energy coherence channels and leave the remaining channels free. With 0.1M trainable parameters, SphereGate outperforms parameter-efficient baselines by 2.0--7.3 points on MATH-500 across Qwen2.5 (0.5B--7B) and Llama-3-8B, is comparable or exceeds full-model GRPO. Anisotropy is not a defect but a division of labor that post-training can exploit.

Low-rank decomposition has emerged as an important problem in Large Language Model (LLM) fine-tuning and inference. Through Singular Value Decomposition (SVD), the weight matrix can be factorized into low-rank spaces optimally. Previously, a common practice was to decompose the weight in the activation-whitened space, and then achieve satisfying results. In this work, we propose Optimal Brain Decomposition LLM (OBD-LLM), which studies the decomposition problem in the model space by utilizing second-order Hessian information. Through a rigorous Kronecker-factorization of the Hessian, we show that the decomposition needs to consider both input and output information of the layer, and achieves much better decomposition results compared to input only method. Our loss-aware decomposition method involves a bi-directional whitening on the weight matrix. As a result, OBD-LLM is a closed-form solution for the optimal decomposition of weights in the language model. Remarkably, we achieve ~20-40% better results than previous state-of-the-art decomposition methods, the SVD-LLM.

Post-training has become central to turning pretrained large language models (LLMs) into aligned, capable, and deployable systems. Recent progress spans supervised fine-tuning (SFT), preference optimization, reinforcement learning (RL), process supervision, verifier-guided methods, distillation, and multi-stage pipelines. Yet these methods are often discussed in fragmented ways, organized by labels or objectives rather than by the behavioral bottlenecks they address. This survey argues that LLM post-training is best understood as structured intervention on model behavior. We organize the field first by trajectory provenance, which defines two primary regimes: off-policy learning on externally supplied trajectories and on-policy learning on learner-generated rollouts. We then interpret methods through two recurring roles -- effective support expansion, which makes useful behaviors more reachable, and policy reshaping, which improves behavior within already reachable regions -- together with a complementary systems-level role, behavioral consolidation, which preserves, transfers, and amortizes useful behavior across stages and model transitions. Under this view, SFT may serve either support expansion or policy reshaping; preference optimization is usually off-policy reshaping, though online variants move closer to learner-generated states. On-policy RL often improves behavior on learner-generated states, but stronger guidance can also make hard-to-reach reasoning paths reachable. Distillation is often better understood as consolidation rather than only compression, and hybrid pipelines emerge as coordinated multi-stage compositions. Overall, the framework helps diagnose post-training bottlenecks and reason about stage composition, suggesting that progress increasingly depends on coordinated systems design rather than any single dominant objective.