We consider self-supervised contrastive learning from unlabeled data. How can we learn good representation that maximizely preserves the semantic structure of embeddings ?

False negative data point (e.g., $x_3^-$, which is a positive sample) should not be pushed apart from the anchor, leading to the true principle.

True negative data point (e.g., $x_1^-$, which is a negative sample) should be pushed apart from the anchor, leading to the hard principle.

• (a) For an given anchor, let $\hat{x}$ be a random variable representing the similarity score between the anchor point and unlabeled samples. We assume that $\hat{x}$ is independently and identically distributed with an unknown distribution.

• (b) Let $\tau^+$ be the class prior probability that an unlabeled sample shares the same latent class as the anchor point (positive class).

• (c) Given an encoder, let $\alpha$ be the probability that the similarity score of a positive sample is higher than that of a negative sample.

BCL uses random samples from the unlabeled data while correcting the resulting bias with importance weights, adhering to the aforementioned true principle and hard principle. Under the problem settings described above, BCL provides an asymptotical unbiased estimation of the supervised loss, and posterior probability estimation of samples being true negatives.