Test Error Guarantees for Batch-normalized two-layer ReLU Networks Trained with Gradient Descent
Keywords: gradient descent, stochastic gradient descent, normalization layers, generalization bounds, margins
TL;DR: In this paper, we provide low training error and test error guarantees of gradient descent (GD) and stochastic gradient descent (SGD) on two-layer ReLU networks with Batch Norm using margin based techniques.
Abstract: This work establishes low training and test error guarantees of gradient descent (GD)
and stochastic gradient descent (SGD) on two-layer ReLU networks with Batch Norm.
Prior work provided convergence analyses for low training error or stationary points
while critically relying on modifications to the setting such as modifying Batch Norm
and assuming the objective is smooth. Although
smoothness based analyses can handle deeper networks, the smoothness constants
are highly non-negligible. We take an alternative approach using
a margin $\gamma$ tailored to normalized networks. In particular, for a
network of width $m$, the
test errors for GD and SGD decrease
at a rate of $O(\frac{m^{1/3}}{\gamma^{1/3} t})$ and $O(\frac{1}{\gamma^2 t})$
up until $t \approx O(\frac {\exp(\gamma^2 m)} n)$. Along the way,
we show that $\gamma$ can be $O(\sqrt{d})$ times larger than
the margin of the max margin linear predictor which can
potentially explain the training and test error speed up for normalized networks.
Supplementary Material: zip
Primary Area: optimization
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Submission Number: 2741
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