Benefits of Additive Noise in Composing Classes with Bounded Capacity
This addresses a foundational issue in modular design for machine learning, offering a general recipe to prevent capacity explosion in composed function classes.
The paper tackles the problem of controlling the capacity of composition classes in machine learning, which can become prohibitively large, by showing that adding a small amount of Gaussian noise effectively manages this capacity, with empirical results on MNIST indicating negligible noise requirements (e.g., standard deviation 10^{-240}).
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.