Statistics Seminar: Alireza Fathollah Pour - Benefits of Additive Noise in Composing Classes with Bounded Capacity
Speaker: Alireza Fathollah Pour, McMaster University
Title: Benefits of Additive Noise in Composing Classes with Bounded Capacity
Abstract: We observe that given two (compatible) classes of functions F and H with small capacity as measured by their uniform covering numbers, the capacity of the composition class can become prohibitively large or even unbounded. We then show that adding a small amount of Gaussian noise to the output of F before composing it with H can effectively control the capacity of their composition, 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. As an extension of our work, we show that adding Gaussian noise to the output of layers in recurrent neural networks can improve the sample complexity of PAC learning them from a linear dependence on input length to a logarithmic dependence.
Date/Time: Tuesday January 31, 2023, 3:30 - 5:00
Location: MDCL 1115