Over-parameterization: A Necessary Condition for Models that Extrapolate

Abstract: In this work, we study over-parameterization as a necessary condition for having the ability for the models to extrapolate outside the convex hull of training set. We specifically, consider classification models, e.g., image classification and other applications of deep learning. Such models are classification functions that partition their domain and assign a class to each partition \cite{strang2019linear}. Partitions are defined by decision boundaries and so is the classification model/function. Convex hull of training set may occupy only a subset of the domain, but trained model may partition the entire domain and not just the convex hull of training set. This is important because many of the testing samples may be outside the convex hull of training set and the way in which a model partitions its domain outside the convex hull would be influential in its generalization. Using approximation theory, we prove that over-parameterization is a necessary condition for having control over the partitioning of the domain outside the convex hull of training set. We also propose a more clear definition for the notion of over-parametrization based on the learning task and the training set at hand. We present empirical evidence about geometry of datasets, both image and non-image, to provide insights about the extent of extrapolation performed by the models. We consider a 64-dimensional feature space learned by a ResNet model and investigate the geometric arrangements of convex hulls and decision boundaries in that space. We also formalize the notion of extrapolation and relate it to the scope of the model. Finally, we review the rich extrapolation literature in pure and applied mathematics, e.g., the Whitney's Extension Problem, and place our theory in that context.