Poster

Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models

Zitong Yang · Yu Bai · Song Mei

Keywords: [ Deep Learning Theory ]

[ Abstract ]
[ Paper ]
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Wed 21 Jul 9 a.m. PDT — 11 a.m. PDT
 
Spotlight presentation: Deep Learning Theory 2
Wed 21 Jul 5 a.m. PDT — 6 a.m. PDT

Abstract: Recent work showed that there could be a large gap between the classical uniform convergence bound and the actual test error of zero-training-error predictors (interpolators) such as deep neural networks. To better understand this gap, we study the uniform convergence in the nonlinear random feature model and perform a precise theoretical analysis on how uniform convergence depends on the sample size and the number of parameters. We derive and prove analytical expressions for three quantities in this model: 1) classical uniform convergence over norm balls, 2) uniform convergence over interpolators in the norm ball (recently proposed by~\citet{zhou2021uniform}), and 3) the risk of minimum norm interpolator. We show that, in the setting where the classical uniform convergence bound is vacuous (diverges to $\infty$), uniform convergence over the interpolators still gives a non-trivial bound of the test error of interpolating solutions. We also showcase a different setting where classical uniform convergence bound is non-vacuous, but uniform convergence over interpolators can give an improved sample complexity guarantee. Our result provides a first exact comparison between the test errors and uniform convergence bounds for interpolators beyond simple linear models.

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