  Fine-tuning is a common practice in deep learning, achieving excellent
generalization results on downstream tasks using relatively little training
data. Although widely used in practice, it is lacking strong theoretical
understanding. We analyze the sample complexity of this scheme for regression
with linear teachers in several architectures. Intuitively, the success of
fine-tuning depends on the similarity between the source tasks and the target
task, however measuring it is non trivial. We show that a relevant measure
considers the relation between the source task, the target task and the
covariance structure of the target data. In the setting of linear regression,
we show that under realistic settings a substantial sample complexity reduction
is plausible when the above measure is low. For deep linear regression, we
present a novel result regarding the inductive bias of gradient-based training
when the network is initialized with pretrained weights. Using this result we
show that the similarity measure for this setting is also affected by the depth
of the network. We further present results on shallow ReLU models, and analyze
the dependence of sample complexity there on source and target tasks. We
empirically demonstrate our results for both synthetic and realistic data.
