Weight Tying Biases Token Embeddings Towards the Output Space

Pretraining of large language models is not only expensive but also prone to certain training instabilities. A specific instability that often occurs at the end of training is output logit divergence. The most widely used mitigation strategies, z-loss and logit soft-capping, merely address the symptoms rather than the underlying cause of the problem. In this paper, we analyze the instability from the perspective of the output embeddings' geometry and identify anisotropic embeddings as its source. Based on this, we propose output embedding centering (OEC) as a new mitigation strategy, and demonstrate that it suppresses output logit divergence. OEC can be implemented in two different ways: as a deterministic operation called $μ$-centering, or a regularization method called $μ$-loss. Our experiments show that both variants outperform z-loss in terms of training stability, while being on par with logit soft-capping. This holds true both in the presence and the absence of weight tying. As a secondary result, we find that $μ$-loss is significantly less sensitive to regularization hyperparameter tuning than z-loss.

The last layer of neural language models (LMs) projects output features of dimension $D$ to logits in dimension $V$, the size of the vocabulary, where usually $D \ll V$. This mismatch is known to raise risks of limited expressivity in neural LMs, creating a so-called softmax bottleneck. We show the softmax bottleneck is not only an expressivity bottleneck but also an optimization bottleneck. Backpropagating $V$-dimensional gradients through a rank-$D$ linear layer induces unavoidable compression, which alters the training feedback provided to the vast majority of the parameters. We present a theoretical analysis of this phenomenon and measure empirically that 95-99% of the gradient norm is suppressed by the output layer, resulting in vastly suboptimal update directions. We conduct controlled pretraining experiments showing that the gradient bottleneck makes trivial patterns unlearnable, and drastically affects the training dynamics of LLMs. We argue that this inherent flaw contributes to training inefficiencies at scale independently of the model architecture, and raises the need for new LM head designs.

Token embeddings are the basic representational units that connect discrete tokens with continuous computation in language models. Although modern language models learn embeddings from random initialization through gradient-based training, the dynamical mechanism by which meaningful embedding structures emerge remains unclear. In this work, we identify that the evolving embedding structures are closely related to token-conditioned label and contextual distributions, which we formalize as probability signatures. We observe a progressive learning process, which we term Context Staircase: embeddings learn the low-order statistic signatures of the data before the high-order ones. More specifically, we observe that early in training they align with the simplest, context-free signature linking a token to its label, and as training proceeds, they progressively reflect signatures involving more and more context tokens. We then analyze the gradient flow of embeddings under small initialization to explain this phenomenon, deriving embedding evolution equations for feed-forward and self-attention architectures. We further extend these observations to real language-model training. Finally, we show that these embedding structures play an important role in both task learning and the incorporation of semantic structure into the embedding space. Overall, our results provide a dynamic explanation of how data statistics and architecture jointly shape token embeddings in language models, and reveal an implicit bias in the space of data statistics: training proceeds from simpler, low-order statistical relations toward increasingly complex, context-dependent ones.