Smoothing the Landscape Boosts the Signal for SGD Optimal Sample Complexity for Learning Single Index Models
We focus on the task of learning a single index model σ(w* · x) with respect to the isotropic Gaussian distribution in d dimensions. Prior work has shown that the sample complexity of learning w* is governed by the information exponent k* of the link function σ, which is defined as the index of the first nonzero Hermite coefficient of σ. Ben Arous et al. [1] showed that n ≳ dk*−1 samples suffice for learning w* and that this is tight for online SGD. However, the CSQ lower bound for gradient based methods only shows that n ≳ dk*/2 samples are necessary. In this work, we close the gap between the upper and lower bounds by showing that online SGD on a smoothed loss learns w* with n ≳ dk*/2 samples. We also draw connections to statistical analyses of tensor PCA and to the implicit regularization effects of minibatch SGD on empirical losses.
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- 4611 Machine learning
- 1702 Cognitive Sciences
- 1701 Psychology
Citation
Published In
ISSN
Publication Date
Volume
Related Subject Headings
- 4611 Machine learning
- 1702 Cognitive Sciences
- 1701 Psychology