  This paper studies few-shot learning via representation learning, where one
uses $T$ source tasks with $n_1$ data per task to learn a representation in
order to reduce the sample complexity of a target task for which there is only
$n_2 (\ll n_1)$ data. Specifically, we focus on the setting where there exists
a good \emph{common representation} between source and target, and our goal is
to understand how much of a sample size reduction is possible. First, we study
the setting where this common representation is low-dimensional and provide a
fast rate of $O\left(\frac{\mathcal{C}\left(\Phi\right)}{n_1T} +
\frac{k}{n_2}\right)$; here, $\Phi$ is the representation function class,
$\mathcal{C}\left(\Phi\right)$ is its complexity measure, and $k$ is the
dimension of the representation. When specialized to linear representation
functions, this rate becomes $O\left(\frac{dk}{n_1T} + \frac{k}{n_2}\right)$
where $d (\gg k)$ is the ambient input dimension, which is a substantial
improvement over the rate without using representation learning, i.e. over the
rate of $O\left(\frac{d}{n_2}\right)$. This result bypasses the
$\Omega(\frac{1}{T})$ barrier under the i.i.d. task assumption, and can capture
the desired property that all $n_1T$ samples from source tasks can be
\emph{pooled} together for representation learning. Next, we consider the
setting where the common representation may be high-dimensional but is
capacity-constrained (say in norm); here, we again demonstrate the advantage of
representation learning in both high-dimensional linear regression and neural
network learning. Our results demonstrate representation learning can fully
utilize all $n_1T$ samples from source tasks.
