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.

}, keywords = {deep learning, generalization error, interpolatory approximation}, issn = {08936080}, doi = {10.1016/j.neunet.2019.08.028}, url = {https://www.sciencedirect.com/science/article/abs/pii/S0893608019302552}, author = {Hrushikesh Mhaskar and Tomaso Poggio} } @article {4240, title = {An analysis of training and generalization errors in shallow and deep networks}, year = {2019}, month = {05/2019}, abstract = {This paper is motivated by an open problem around deep networks, namely, the apparent absence of overfitting 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.

}, keywords = {deep learning, generalization error, interpolatory approximation}, author = {Hrushikesh Mhaskar and Tomaso Poggio} } @article {3315, title = {An analysis of training and generalization errors in shallow and deep networks}, year = {2018}, month = {02/2018}, abstract = {An open problem around deep networks is the apparent absence of over-fitting despite large over-parametrization which allows perfect fitting of the training data. In this paper, we explain this phenomenon when each unit evaluates a trigonometric polynomial. It is well understood in the theory of function approximation that ap- proximation by trigonometric polynomials is a {\textquotedblleft}role model{\textquotedblright} for many other processes of approximation that have inspired many theoretical constructions also in the context of approximation by neural and RBF networks. In this paper, we argue that the maximum loss functional is necessary to measure the generalization error. We give estimates on exactly how many parameters ensure both zero training error as well as a good generalization error, and how much error to expect at which test data. An interesting feature of our new method is that the variance in the training data is no longer an insurmountable lower bound on the generalization error.

}, keywords = {deep learning, generalization error, interpolatory approximation}, author = {Hrushikesh Mhaskar and Tomaso Poggio} }