DEEP LEARNING GENERALIZES BECAUSE THE PARAMETER-FUNCTION MAP IS BIASED TOWARDS SIMPLE FUNCTIONS
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 bias in a model DNN for Boolean functions, as well as in much larger fully conected and convolutional networks trained on 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 CIFAR10and for architectures including convolutional and fully connected networks.
The authors make a case that deep networks are biased
toward fitting data with simple functions.
The start by examining the priors on classifiers obtained by sampling
the weights of a neural network according to different distributions.  They do this
in two ways.  First, they examine properties of the distribution
on binary-valued functions on seven boolean inputs obtained by
sampling the weights of a small neural network.  They also empirically compare
the labelings obtained by sampling the weights of a network with
labelings obtained from a Gaussian process model arising from earlier
work.
Next, they analyze the complexity of the functions produced, using
different measures of the complexity of boolean functions.  A
favorite of theirs is something that they call Lempel-Ziv complexity,
which is measured by choosing an arbitrarily ordering of the
domain, writing the outputs of the function in that ordering,
and looking at how well the Lempel-Ziv algorithm compresses this
sequence.  I am not convinced that this is the most meaningful
and fundamental measure of the complexity of functions.
(In the supplementary material, they examine some others.
They show plots relating the different measures in the body
of the paper.  None of the measures is specified in detail in the
body of the paper. They provide plots relating these complexity
measures, but they don't demonstrate a very close connection.)
The authors then evaluate the generalization bound obtained by
applying a PAC Bayes bound, together with the assumption that
the training process produces weights sampled from the distribution
obtained by conditioning weights chosen according to the random
initialization on the event that they fit they fit the training
data perfectly.  They do this for small networks and simple datasets.
They bounds are loose, but not vacuous, and follow the same order
of difficulty on a handful of datasets as the true generalization
error.
In all of their experiments, they stop training when the training
accuracy reaches 100%, where papers like https://arxiv.org/pdf/1706.08947.pdf
have found that continuing training past this point further improves test
accuracy.  The experiments all use architectures that are
quite dissimilar to what is commonly used in practice, and
achieve much worse accuracy, so that a reader is concerned
that the results differ qualitatively in other respects.
I do not find it surprising that randomly sampling parameters
of deep networks leads to simple functions.
Papers like the Soudry, et al paper cited in this submission are
inconsistent with the assumption in the paper that SGD samples
parameters uniformly.
It is not clear to me how many hidden layers were used for the
results in Table 1 (is it four?).  
I did find it interesting to see exactly how concentrated the
distribution of functions obtained in their 7-input experiment
was, and also found results on the agreement of the Gaussian process
models with the randomly sampled weight interesting, as far as they
went.  Overall, I am not sure that this paper provided enough
fundamental new insight to be published in ICLR.