We study the language-model head and softmax as a single module, deriving an update geometry from their composition rather than from the weight matrix in isolation. Under Hilbert's projective distance, the maximum change caused by an update $S$ over $\left|\left|{h}\right|\right|_2\le H$ is $H\max_{i<j}\left|\left|{s_i-s_j}\right|\right|_2$, which is $H$ times the Euclidean diameter of its token rows. Motivated by Muon's singular-value conditioning, we propose maximizing the smallest row separation while constraining this diameter, producing an approximate-equidistance problem when $V\gg d$.
The language-model head maps a hidden state of width D to a vocabulary of size V, so its transpose can return at most D independent directions to the Transformer. Godey and Artzi argue that this severe projection is a harmful optimization bottleneck. We separate the geometry from the causal claim. Our backward-only intervention keeps the ordinary logits and the exact LM-head parameter update while reducing only the rank of the gradient sent into the Transformer. Across five paired seeds on byte-level and BPE-8192 WikiText-2 models, reducing backward rank increases validation loss. An equally ranked factorized forward head, however, increases loss substantially more. At half rank in the larger model, the backward-only loss increase is 0.0586 (95% CI [0.0167, 0.1005]), while the factorized forward head increases loss by 0.1795 ([0.1547, 0.2042]). The vocabulary-space residual also contributes to the ordinary LM-head update, and removing that contribution is harmful. Additional controls show that repeated-token failures are confounded by the number of independently sampled symbols, that adding never-target output classes does not impair learning, and that projection diagnostics do not reliably predict progress in our runs. Tested auxiliary feedback routes do not beat tuned backpropagation. These results confirm strong geometric compression but do not establish that it is a harmful optimization bottleneck.
The normalized Transformer (nGPT) realizes hyperspherical representation learning by constraining model parameter vectors and activation vectors to the unit hypersphere. In this paper, we describe a practical training recipe for nGPT and evaluate it on modern hybrid Mamba-2--Transformer Mixture-of-Experts (MoE) models. The recipe introduces Logit Gradient Preconditioning, Logarithmic Learning Rate Decay, GatedAdamW, angular update control, and optional exploration mechanisms. Compared with an unnormalized model of the same hybrid MoE architecture trained with AdamW, the 30B-total-parameter nGPT model reaches the same validation loss using approximately half as many training tokens. The recipe scales across the models considered, which contain up to 30B total parameters.
Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood. This preliminary report proposes a unified framework built on a single idealizing assumption -- exact scale invariance of the loss under weight rescaling, which holds approximately in normalization-heavy networks. Under this assumption, plain SGD carries a built-in 1/||W|| brake on its update size, whereas Muon's matrix-sign step removes that brake, so both the Frobenius and spectral norms drift outward faster (t^{1/2} versus t^{1/4}). We further observe that the spectral-norm perturbation has a non-negative second-order term. This implies that a lightweight "spectral cap" -- which projects out only the first-order growth of the single top singular direction from each update -- can control the output covariance W K_X W^T without freezing training: the weight keeps learning through non-top directions, top-direction rotation, and top switching. We relate this cap to the min-entropy (H-infinity) of the singular-value spectrum. We then study three systems trained with Muon: a nanoGPT feed-forward projection, a 64-expert mixture-of-experts router, and the query/key projections of a bf16 FlashAttention block. In each case the cap increases isotropy and, at the margins -- a router collapsing to a single expert, and the near-divergence of one attention head -- prevents a concrete failure, while leaving validation loss essentially unchanged. We emphasize that the scale-invariance assumption is strong and that these small-scale results are preliminary; comments are welcome.
On-Policy Self-Distillation (OPSD) has emerged as a crucial paradigm for enhancing and aligning Large Language Models (LLMs). However, in complex reasoning tasks, OPSD paradoxically degrades downstream performance. In this paper, we systematically investigate this pathology and identify a severe optimization trap we define as \textbf{Thinking Collapse} -- a sharp decline in the model's native intermediate reasoning behavior, measured by epistemic-token density (ET per 1k). Through entropy-based gradient masking and token-level target analysis, we show that this collapse is triggered by aggressive teacher gradients at high-student-entropy decision forks, where student epistemic tokens are frequently suppressed into teacher non-epistemic targets and are highly concentrated in high pointwise student-teacher divergence regions. To resolve this optimization pathology, we propose \textbf{Adaptive Dual-Perspective OPSD (AD-OPSD)}, a robust control framework that dynamically moderates the self-distillation objective. AD-OPSD selectively anchors high-suppression-risk sandboxed tokens to a reference prior derived from the frozen base model via an asymmetrical pointwise divergence gate, preserving native thinking capacity while retaining OPSD's error-correcting power. Extensive experiments across competitive mathematical benchmarks show that AD-OPSD improves over standard OPSD by up to \textbf{+4.1\%} absolute average accuracy across diverse model scales and datasets. Further analysis demonstrates that AD-OPSD mitigates thinking collapse and generalizes robustly to different post-training paradigms.
Many modern Language Model (LM) pipelines return an averaged model, such as an exponential moving average of the training iterates, rather than the final iterate itself. This raises a fundamental question: given that we will return an iterate average, how should we change training to improve the performance of this average? We study this question by formulating optimizer design for the iterate-average estimator as an optimal-control problem. In a continuous-time stochastic quadratic model, we solve for the control strategy that minimizes the error of the returned average subject to a penalty on the size of the intervention. A practical approximation to this controller yields PACE, a lightweight wrapper around AdamW that pulls the live weights toward their exponential moving average with a clipped, per-coordinate control strength. We prove that a stylized version of PACE converges at the standard stochastic convex optimization rate, up to a factor depending on the averaging rule, while in the quadratic setting it can strictly improve the limiting squared error of the iterate-average estimator and can do so by an arbitrarily large factor on some instances. Empirically, our results suggest that PACE improves over AdamW and EMA-evaluated AdamW in supervised fine-tuning of 1-2B parameter LMs and in GPT-2 pretraining on FineWeb for a wide range of learning rates, decay schedules, and other hyperparameters.
On-policy distillation (\textsc{OPD}) has recently become a prominent post-training recipe as it combines two desirable ingredients: on-policy student trajectories and dense teacher supervision, yet how this hybrid changes a model's parameters remains unclear. Across several language and vision-language model pairs and use cases, our analysis yields two main findings. On sparsity, \textsc{OPD}-style updates are small and coordinate-sparse. They are distributed across layers and are usually FFN-heavy. This sparse structure is operationally useful: training only the discovered subnetwork recovers nearly the same performance as full \textsc{OPD}. However, the sparsity-inducing SGD optimizer underperforms AdamW in our optimizer ablation, likely because dense teacher supervision preserves heterogeneous coordinate-wise gradient scales where AdamW's adaptive scaling remains useful. On geometry, the updates are numerically full-rank but spectrally concentrated; they lie mostly away from the principal singular subspaces of the source weights and fall disproportionately on coordinates where the source weights are close to zero. These findings suggest that dense teacher supervision does not turn \textsc{OPD} into ordinary dense parameter rewriting; instead, \textsc{OPD} retains important geometric signatures of on-policy post-training.
In Low-Rank Adaptation (LoRA), the scaling factor $α$ is often treated as a mere complement to the learning rate, yet its role in optimization remains poorly understood. In this paper, we reveal that the scaling factor $α$ and the learning rate function differently, with $α$ emerging as the dominant driver of effective optimization, delivering gains that cannot be replicated by learning rate scaling alone. Through the synergy of extensive empirical analysis and a theoretical Signal-Drift framework, we uncover three findings into LoRA's scaling mechanism: First, LoRA's spectral suppression smooths the optimization landscape, rendering standard hyperparameters overly conservative and creating an optimization gap. Second, when leveraging this smoothness to accelerate convergence, $α$ outperforms the learning rate by amplifying the task signal without increasing the drift ratio. Third, the optimal scaling factor follows a sublinear relationship with the rank, well characterized by a square-root law with an unexpectedly large coefficient, revealing the insufficient scaling of existing rank-tied heuristics. Based on these insights, we propose LoRA-$α$, a minimalist framework that restores $α$ to its principled regime, making LoRA compatible with standard small learning rates. Extensive evaluations across diverse tasks demonstrate that LoRA-$α$ consistently improves performance while streamlining hyperparameter search, unleashing the learning potential of LoRA.