  This paper focuses on understanding how the generalization error scales with
the amount of the training data for deep neural networks (DNNs). Existing
techniques in statistical learning require computation of capacity measures,
such as VC dimension, to provably bound this error. It is however unclear how
to extend these measures to DNNs and therefore the existing analyses are
applicable to simple neural networks, which are not used in practice, e.g.,
linear or shallow ones or otherwise multi-layer perceptrons. Moreover, many
theoretical error bounds are not empirically verifiable. We derive estimates of
the generalization error that hold for deep networks and do not rely on
unattainable capacity measures. The enabling technique in our approach hinges
on two major assumptions: i) the network achieves zero training error, ii) the
probability of making an error on a test point is proportional to the distance
between this point and its nearest training point in the feature space and at a
certain maximal distance (that we call radius) it saturates. Based on these
assumptions we estimate the generalization error of DNNs. The obtained estimate
scales as O(1/(\delta N^{1/d})), where N is the size of the training data and
is parameterized by two quantities, the effective dimensionality of the data as
perceived by the network (d) and the aforementioned radius (\delta), both of
which we find empirically. We show that our estimates match with the
experimentally obtained behavior of the error on multiple learning tasks using
benchmark data-sets and realistic models. Estimating training data requirements
is essential for deployment of safety critical applications such as autonomous
driving etc. Furthermore, collecting and annotating training data requires a
huge amount of financial, computational and human resources. Our empirical
estimates will help to efficiently allocate resources.
