  Through minimization of an appropriate loss function such as the InfoNCE
loss, contrastive learning (CL) learns a useful representation function by
pulling positive samples close to each other while pushing negative samples far
apart in the embedding space. The positive samples are typically created using
"label-preserving" augmentations, i.e., domain-specific transformations of a
given datum or anchor. In absence of class information, in unsupervised CL
(UCL), the negative samples are typically chosen randomly and independently of
the anchor from a preset negative sampling distribution over the entire
dataset. This leads to class-collisions in UCL. Supervised CL (SCL), avoids
this class collision by conditioning the negative sampling distribution to
samples having labels different from that of the anchor. In hard-UCL (H-UCL),
which has been shown to be an effective method to further enhance UCL, the
negative sampling distribution is conditionally tilted, by means of a hardening
function, towards samples that are closer to the anchor. Motivated by this, in
this paper we propose hard-SCL (H-SCL) {wherein} the class conditional negative
sampling distribution {is tilted} via a hardening function. Our simulation
results confirm the utility of H-SCL over SCL with significant performance
gains {in downstream classification tasks.} Analytically, we show that {in the}
limit of infinite negative samples per anchor and a suitable assumption, the
{H-SCL loss} is upper bounded by the {H-UCL loss}, thereby justifying the
utility of H-UCL {for controlling} the H-SCL loss in the absence of label
information. Through experiments on several datasets, we verify the assumption
as well as the claimed inequality between H-UCL and H-SCL losses. We also
provide a plausible scenario where H-SCL loss is lower bounded by UCL loss,
indicating the limited utility of UCL in controlling the H-SCL loss.
