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Poster
A Sample Complexity Separation between Non-Convex and Convex Meta-Learning
Nikunj Umesh Saunshi · Yi Zhang · Mikhail Khodak · Sanjeev Arora

Tue Jul 14 10:00 AM -- 10:45 AM & Tue Jul 14 09:00 PM -- 09:45 PM (PDT) @ None #None
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 {\em 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 {\em 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 $O(1)$, 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.

Author Information

Nikunj Saunshi (Princeton University)
Yi Zhang (Princeton University)
Misha Khodak (CMU)
Sanjeev Arora (Princeton University and Institute for Advanced Study)

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