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\inft