An analysis of training and generalization errors in shallow and deep networks

TL;DR

This paper investigates why deep networks often do not overfit despite over-parametrization, analyzing the relationship between training and generalization errors in shallow and deep networks with periodic activations.

Contribution

It introduces a new perspective on measuring generalization error using maximum loss and provides estimates on parameter counts for zero training and good generalization.

Findings

Maximum loss is a better measure of generalization in compositional functions.

Regularization solutions can guarantee both low training and generalization errors.

Estimates of parameters needed for perfect fitting and good generalization are provided.

Abstract

This paper is motivated by an open problem around deep networks, namely, the apparent absence of over-fitting despite large over-parametrization which allows perfect fitting of the training data. In this paper, we analyze this phenomenon in the case of regression problems when each unit evaluates a periodic activation function. We argue that the minimal expected value of the square loss is inappropriate to measure the generalization error in approximation of compositional functions in order to take full advantage of the compositional structure. Instead, we measure the generalization error in the sense of maximum loss, and sometimes, as a pointwise error. We give estimates on exactly how many parameters ensure both zero training error as well as a good generalization error. We prove that a solution of a regularization problem is guaranteed to yield a good training error as well as a good generalization error and estimate how much error to expect at which test data.