Benefits of Additive Noise in Composing Classes with Bounded CapacityDownload PDF

Published: 31 Oct 2022, Last Modified: 22 Oct 2023NeurIPS 2022 AcceptReaders: Everyone
Keywords: Generalization Bound, Composition, Noise, Covering Number, Neural Networks
TL;DR: We develop a theory showing that adding a little bit of noise can effectively control the capacity of composite classes.
Abstract: We observe that given two (compatible) classes of functions $\mathcal{F}$ and $\mathcal{H}$ with small capacity as measured by their uniform covering numbers, the capacity of the composition class $\mathcal{H} \circ \mathcal{F}$ can become prohibitively large or even unbounded. We then show that adding a small amount of Gaussian noise to the output of $\mathcal{F}$ before composing it with $\mathcal{H}$ can effectively control the capacity of $\mathcal{H} \circ \mathcal{F}$, offering a general recipe for modular design. To prove our results, we define new notions of uniform covering number of random functions with respect to the total variation and Wasserstein distances. We instantiate our results for the case of multi-layer sigmoid neural networks. Preliminary empirical results on MNIST dataset indicate that the amount of noise required to improve over existing uniform bounds can be numerically negligible (i.e., element-wise i.i.d. Gaussian noise with standard deviation $10^{-240}$)
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