  One popular trend in meta-learning is to learn from many training tasks a
common initialization for a gradient-based method that can be used to solve a
new task with few samples. The theory of meta-learning is still in its early
stages, with several recent learning-theoretic analyses of methods such as
Reptile [Nichol et al., 2018] being for convex models. This work shows that
convex-case analysis might be insufficient to understand the success of
meta-learning, and that even for non-convex models it is important to look
inside the optimization black-box, specifically at properties of the
optimization trajectory. We construct a simple meta-learning instance that
captures the problem of one-dimensional subspace learning. For the convex
formulation of linear regression on this instance, we show that the new task
sample complexity of any initialization-based meta-learning algorithm is
$\Omega(d)$, where $d$ is the input dimension. In contrast, for the non-convex
formulation of a two layer linear network on the same instance, we show that
both Reptile and multi-task representation learning can have new task sample
complexity of $\mathcal{O}(1)$, demonstrating a separation from convex
meta-learning. Crucially, analyses of the training dynamics of these methods
reveal that they can meta-learn the correct subspace onto which the data should
be projected.
