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Learning operators between infinitely dimensional spaces is an important learning task arising in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric estimation of Lipschitz operators using deep neural networks. Non-asymptotic upper bounds are derived for the generalization error of the empirical risk minimizer over a properly chosen network class. Under the assumption that the target operator exhibits a low dimensional structure, our error bounds decay as the training sample size increases, with an attractive fast rate depending on the intrinsic dimension in our estimation. Our assumptions cover most scenarios in real applications and our results give rise to fast rates by exploiting low dimensional structures of data in operator estimation. We also investigate the influence of network structures (e.g., network width, depth, and sparsity) on the generalization error of the neural network estimator and propose a general suggestion on the choice of network structures to maximize the learning efficiency quantitatively.
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Maximal Initial Learning Rates in Deep ReLU Networks
Training a neural network requires choosing a suitable learning rate, which involves a trade-off between speed and effectiveness of convergence. While there has been considerable theoretical and empirical analysis of how large the learning rate can be, most prior work focuses only on late-stage training. In this work, we introduce the maximal initial learning rate - the largest learning rate at which a randomly initialized neural network can successfully begin training and achieve (at least) a given threshold accuracy. Using a simple approach to estimate the maximal initial learning rate, we observe that in constant-width fully-connected ReLU networks, the maximal initial learning rate behaves differently from the maximum learning rate later in training. Specifically, we find that the maximal initial learning rate is well predicted as a power of depth times width, provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer is trained at a relatively small learning rate. We further analyze the relationship between the maximal initial learning rate and the sharpness of the network at initialization, indicating they are closely though not inversely related. We formally prove bounds for the maximal initial learning rate in terms of depth times width that align with our empirical results.
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- PAR ID:
- 10479889
- Publisher / Repository:
- Proceedings of Machine Learning Research
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- ISSN:
- 2640-3498
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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