Optimization Dynamics Imprint Semantic Specificity in Contrastive Embedding Norms

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.

Multimodal embedding models encode heterogeneous inputs into a shared embedding space, enabling efficient similarity computation across modalities and tasks. Most existing methods optimize cosine-based contrastive objectives, which promote stable training but restrict semantic compatibility to angular geometry, precluding embedding norms from serving as an additional semantic signal. However, directly optimizing the more expressive dot-product similarity, which leverages both angular and norm information, underperforms cosine-based training and exhibits unstable training dynamics. We attribute this discrepancy to premature optimization-space expansion, manifested as angular--norm entanglement and directional anisotropy in the representation space and further compounded by full-parameter fine-tuning. In this paper, we propose PACE, a two-stage framework that progressively expands both the representation and trainable parameter spaces. Stage I combines cosine-based objective with low-rank adaptation to establish a reliable angular geometry within constrained optimization spaces. Stage II switches to dot-product similarity and full-parameter fine-tuning, enabling embedding directions and norms to jointly encode semantic information. We further introduce Focal Embedding Loss, a confidence-adaptive objective that downweights queries with high positive retrieval confidence while emphasizing ambiguous queries with competitive negatives. Experiments across multiple backbone scales and diverse multimodal embedding tasks consistently validate the effectiveness of PACE.

Similarity search is a primary application of embedding models trained by contrastive learning. For one of the most popular contrastive learning loss functions, InfoNCE, we show that the population risk with $k$ negative samples is $O(1/k)$ close to an expected cross-entropy which quantifies deviation between i) a softmax similarity search over unseen data using the learned embedding function, and ii) an idealised softmax search over the same data but using similarity implicitly represented in the positive sample generator. This complements existing interpretations of InfoNCE in the $k\to\infty$ limit which are phrased in terms of mutual information, and alignment versus uniformity in embeddings. To quantify generalisation performance, we introduce a new continuity bound for the InfoNCE loss, obtained via Gâteaux differentiation. The bound preserves the structure of averaging over negative samples present in the loss function and features an ``inverse temperature'' parameter which can be tuned to account for the algorithmic temperature. For embedding functions which are Lipschitz in a parameter, this yields a simple demonstration that the averaging effect of $k$ negative samples in the InfoNCE loss carries over to stabilisation of the generalisation error as $k$ grows.