  Deep neural networks (DNNs) generalize remarkably well without explicit
regularization even in the strongly over-parametrized regime where classical
learning theory would instead predict that they would severely overfit. While
many proposals for some kind of implicit regularization have been made to
rationalise this success, there is no consensus for the fundamental reason why
DNNs do not strongly overfit. In this paper, we provide a new explanation. By
applying a very general probability-complexity bound recently derived from
algorithmic information theory (AIT), we argue that the parameter-function map
of many DNNs should be exponentially biased towards simple functions. We then
provide clear evidence for this strong simplicity bias in a model DNN for
Boolean functions, as well as in much larger fully connected and convolutional
networks applied to CIFAR10 and MNIST. As the target functions in many real
problems are expected to be highly structured, this intrinsic simplicity bias
helps explain why deep networks generalize well on real world problems. This
picture also facilitates a novel PAC-Bayes approach where the prior is taken
over the DNN input-output function space, rather than the more conventional
prior over parameter space. If we assume that the training algorithm samples
parameters close to uniformly within the zero-error region then the PAC-Bayes
theorem can be used to guarantee good expected generalization for target
functions producing high-likelihood training sets. By exploiting recently
discovered connections between DNNs and Gaussian processes to estimate the
marginal likelihood, we produce relatively tight generalization PAC-Bayes error
bounds which correlate well with the true error on realistic datasets such as
MNIST and CIFAR10 and for architectures including convolutional and fully
connected networks.
