  We provide new statistical guarantees for transfer learning via
representation learning--when transfer is achieved by learning a feature
representation shared across different tasks. This enables learning on new
tasks using far less data than is required to learn them in isolation.
Formally, we consider $t+1$ tasks parameterized by functions of the form $f_j
\circ h$ in a general function class $\mathcal{F} \circ \mathcal{H}$, where
each $f_j$ is a task-specific function in $\mathcal{F}$ and $h$ is the shared
representation in $\mathcal{H}$. Letting $C(\cdot)$ denote the complexity
measure of the function class, we show that for diverse training tasks (1) the
sample complexity needed to learn the shared representation across the first
$t$ training tasks scales as $C(\mathcal{H}) + t C(\mathcal{F})$, despite no
explicit access to a signal from the feature representation and (2) with an
accurate estimate of the representation, the sample complexity needed to learn
a new task scales only with $C(\mathcal{F})$. Our results depend upon a new
general notion of task diversity--applicable to models with general tasks,
features, and losses--as well as a novel chain rule for Gaussian complexities.
Finally, we exhibit the utility of our general framework in several models of
importance in the literature.
