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Title: Deterministic PAC-Bayesian generalization bounds for deep networks via generalizing noise-resilience

Paper content:
The ability of overparameterized deep networks to generalize well has been linked to the fact that stochastic gradient descent (SGD) finds solutions that lie in flat, wide minima in the training loss -- minima where the output of the network is resilient to small random noise added to its parameters.
So far this observation has been used to provide generalization guarantees only for neural networks whose parameters are either \textit{stochastic} or \textit{compressed}.
In this work, we present a general PAC-Bayesian framework that leverages this observation to provide a bound on the original network learned -- a network that is deterministic and uncompressed.
What enables us to do this is a key novelty in our approach: our framework allows us to show that if on training data, the interactions between the weight matrices satisfy certain conditions that imply a wide training loss minimum, these conditions themselves {\em generalize} to the interactions between the matrices on test data, thereby implying a wide test loss minimum.
We then apply our general framework in a setup where we assume that the pre-activation values of the network are not too small (although we assume this only on the training data).
In this setup, we provide a generalization guarantee for the original (deterministic, uncompressed) network, that does not scale with product of the spectral norms of the weight matrices -- a guarantee that would not have been possible with prior approaches.
Modern deep neural networks contain millions of parameters and are trained on relatively few samples.
Conventional wisdom in machine learning suggests that such models should massively overfit on the training data, as these models have the capacity to memorize even a randomly labeled dataset of similar size (Zhang et al., 2017; Neyshabur et al., 2015) .
Yet these models have achieved state-ofthe-art generalization error on many real-world tasks.
This observation has spurred an active line of research (Soudry et al., 2018; BID2 BID11 ) that has tried to understand what properties are possessed by stochastic gradient descent (SGD) training of deep networks that allows these networks to generalize well.One particularly promising line of work in this area (Neyshabur et al., 2017; BID0 has been bounds that utilize the noise-resilience of deep networks on training data i.e., how much the training loss of the network changes with noise injected into the parameters, or roughly, how wide is the training loss minimum.
While these have yielded generalization bounds that do not have a severe exponential dependence on depth (unlike other bounds that grow with the product of spectral norms of the weight matrices), these bounds are quite limited: they either apply to a stochastic version of the classifier (where the parameters are drawn from a distribution) or a compressed version of the classifier (where the parameters are modified and represented using fewer bits).In
this paper, we revisit the PAC-Bayesian analysis of deep networks in Neyshabur et al. (2017; and provide a general framework that allows one to use noise-resilience of the deep network on training data to provide a bound on the original deterministic and uncompressed network. We
achieve this by arguing that if on the training data, the interaction between the 'activated weight matrices' (weight matrices where the weights incoming from/outgoing to inactive units are zeroed out) satisfy certain conditions which results in a wide training loss minimum, these conditions themselves generalize to the weight matrix interactions on the test data.After presenting this general PAC-Bayesian framework, we specialize it to the case of deep ReLU networks, showing that we can provide a generalization bound that accomplishes two goals simultaneously: i)
it applies to the original network and ii
) it does not scale exponentially with depth in terms of the products of the spectral norms of the weight matrices; instead our bound scales with more meaningful terms that capture the interactions between the weight matrices and do not have such a severe dependence on depth in practice. We
note that all but one of these terms are indeed quite small on networks in practice. However
, one particularly (empirically) large term that we use is the reciprocal of the magnitude of the network pre-activations on the training data (and so our bound would be small only in the scenario where the pre-activations are not too small). We emphasize
that this drawback is more of a limitation in how we characterize noise-resilience through the specific conditions we chose for the ReLU network, rather than a drawback in our PAC-Bayesian framework itself. Our hope is
that, since our technique is quite general and flexible, by carefully identifying the right set of conditions, in the future, one might be able to derive a similar generalization guarantee that is smaller in practice.To the best of our knowledge, our approach of generalizing noise-resilience of deep networks from training data to test data in order to derive a bound on the original network that does not scale with products of spectral norms, has neither been considered nor accomplished so far, even in limited situations.

assistant: text: We provide a PAC-Bayes based generalization guarantee for uncompressed, deterministic deep networks by generalizing noise-resilience of the network on the training data to the test data.