An analysis of training and generalization errors in shallow and deep networks
This addresses a foundational open problem in deep learning theory for researchers, providing theoretical guarantees on generalization in over-parameterized networks.
The paper tackles the problem of understanding why deep networks avoid overfitting despite over-parametrization, focusing on regression with periodic activation functions, and proves that a regularization solution ensures zero training error and good generalization error with estimates on parameter counts and error bounds.
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.