Similarity search generalisation in contrastive learning with InfoNCE loss

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 infinity 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 G a teaux 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.