When Softmax Fails at the Top: Extreme Value Corrections for InfoNCE

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.

Contrastive learning (CL) aims to preserve relational structure between samples by learning representations that reflect a similarity graph. Yet, the geometry of the resulting embeddings remains poorly understood. Here we show that weighted InfoNCE objectives can be interpreted as Distance Geometry Problems, where the weighting scheme specifies the target geometry to be realized by the representation. This viewpoint yields exact characterizations of the optimal embeddings for several supervised and weakly supervised objectives. In supervised classification, both SupCon and Soft SupCon (a dense relaxation of it where pairs from distinct classes have small non-zero similarity) collapse samples within each class to a single prototype. However, while balanced SupCon recovers the classical regular simplex geometry, class imbalance breaks this symmetry: SupCon induces non-uniform inter-class similarities depending on class sizes, whereas Soft SupCon preserves a regular simplex geometry regardless of class imbalance. In continuous-label settings, our framework reveals a different failure mode: y-Aware CL generally cannot attain its entropic optimum unless the labels lie on a hypersphere, exposing a mismatch between Euclidean label weights and spherical latent similarity. By contrast, geometrically consistent choices such as Euclidean-Euclidean weighting or X-CLR admit unique optimal embeddings. Our results show that the choice of weighting scheme determines whether contrastive learning is geometrically realizable, degenerate, or inconsistent, providing a principled framework for designing contrastive objectives.

Contrastive learning relies on informative negatives to shape the representation space, yet obtaining hard negatives is costly, often requiring large batch sizes or extensive memory banks. We propose SynCo (Synthetic negatives in Contrastive learning), an approach that synthesizes hard negatives directly in the representation space from cached queue embeddings, with no additional forward passes or input-space processing. We find that six lightweight synthesis strategies, exhaustively covering the geometric, stochastic, and adversarial perturbation families, consistently improve learned representations at negligible computational cost. Although applicable to any InfoNCE-based contrastive objective, we demonstrate SynCo within the MoCo framework. On ImageNet ILSVRC-2012 linear evaluation at 200 epochs, SynCo yields improvements of +0.4% over MoCo-v2 and +1.0% over MoCHi. Unlike MoCHi, which degrades at extended pretraining schedules (underperforming MoCo-v2 by 2.4% at 800 epochs), SynCo does not: with a simple synthetic negative schedule, performance improves by +0.5% over MoCo-v2 at 800 epochs. SynCo also transfers well to a range of downstream tasks.