It is currently known how to characterize functions that neural networks can learn with SGD for two extremal parametrizations: neural networks in the linear regime, and neural networks with no structural constraints. However, for the main parametrization of interest —non-linear but regular networks— no tight characterization has yet been achieved, despite significant developments.
We take a step in this direction by considering depth-2 neural networks trained by SGD in the mean-field regime. We consider functions on binary inputs that depend on a latent low-dimensional subspace (i.e., small number of coordinates). This regime is of interest since it is poorly under- stood how neural networks routinely tackle high-dimensional datasets and adapt to latent low- dimensional structure without suffering from the curse of dimensionality. Accordingly, we study SGD-learnability with O(d) sample complexity in a large ambient dimension d.
Our main results characterize a hierarchical property —the merged-staircase property— that is both necessary and nearly sufficient for learning in this setting. We further show that non-linear training is necessary: for this class of functions, linear methods on any feature map (e.g., the NTK) are not capable of learning efficiently. The key tools are a new “dimension-free” dynamics approximation result that applies to functions defined on a latent space of low-dimension, a proof of global convergence based on polynomial identity testing, and an improvement of lower bounds against linear methods for non-almost orthogonal functions.
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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 with respect to the isotropic Gaussian distribution in d dimensions. Prior work has shown that the sample complexity of learning the hidden direction is governed by the information exponent k of the link function. Ben Arous et al. showed that d^k samples suffice for learning and that this is tight for online SGD. However, the CSQ lowerbound for gradient based methods only shows that d^{k/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 the hidden direction with the correct number of 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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- Award ID(s):
- 1845171
- PAR ID:
- 10530894
- Publisher / Repository:
- NeurIPS 2023
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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