Benefits of Additive Noise in Composing Classes with Bounded Capacity

TL;DR

Adding a small amount of Gaussian noise to the output of function classes can significantly control the capacity of their compositions, leading to better modular design and improved bounds, as demonstrated on neural networks and MNIST.

Contribution

The paper introduces a novel approach of adding Gaussian noise to control the capacity of composed function classes, with new notions of uniform covering numbers for random functions.

Findings

Noise can effectively bound the capacity of composed classes.

Empirical results show negligible noise improves bounds on MNIST.

New theoretical tools for analyzing random function capacities.

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 $10^{-240}$). The source codes are available at https://github.com/fathollahpour/composition_noise.