  Given two networks with the same training loss on a dataset, when would they
have drastically different test losses and errors? Better understanding of this
question of generalization may improve practical applications of deep networks.
In this paper we show that with cross-entropy loss it is surprisingly simple to
induce significantly different generalization performances for two networks
that have the same architecture, the same meta parameters and the same training
error: one can either pretrain the networks with different levels of
"corrupted" data or simply initialize the networks with weights of different
Gaussian standard deviations. A corollary of recent theoretical results on
overfitting shows that these effects are due to an intrinsic problem of
measuring test performance with a cross-entropy/exponential-type loss, which
can be decomposed into two components both minimized by SGD -- one of which is
not related to expected classification performance. However, if we factor out
this component of the loss, a linear relationship emerges between training and
test losses. Under this transformation, classical generalization bounds are
surprisingly tight: the empirical/training loss is very close to the
expected/test loss. Furthermore, the empirical relation between classification
error and normalized cross-entropy loss seem to be approximately monotonic
