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We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by neural networks. Such algorithms (most prominently stochastic gradient descent and its variants) are used extensively in the field of deep learning. One of the most important problems in this field concerns the question of whether it is possible to realize theoretically provable neural network approximation rates by such algorithms. We answer this question in the negative by proving hardness results for the problems of approximation and integration on a novel class of neural network approximation spaces. In particular, our results confirm a conjectured and empirically observed theory-to-practice gap in deep learning. We complement our hardness results by showing that approximation rates of a comparable order of convergence are (at least theoretically) achievable.
This paper addresses the growing need to process non-Euclidean data, by introducing a geometric deep learning (GDL) framework for building universal feedforward-type models compatible with differentiable manifold geometries. We show that our GDL mode
Progressive Neural Network Learning is a class of algorithms that incrementally construct the networks topology and optimize its parameters based on the training data. While this approach exempts the users from the manual task of designing and valida
Complex networks are often either too large for full exploration, partially accessible, or partially observed. Downstream learning tasks on these incomplete networks can produce low quality results. In addition, reducing the incompleteness of the net
We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated
Despite recent advances in its theoretical understanding, there still remains a significant gap in the ability of existing PAC-Bayesian theories on meta-learning to explain performance improvements in the few-shot learning setting, where the number o