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Poster

Generalization Bounds for Neural Networks via Approximate Description Length

Amit Daniely · Elad Granot

East Exhibition Hall B, C #228

Keywords: [ Learning Theory ] [ Theory ]


Abstract: We investigate the sample complexity of networks with bounds on the magnitude of its weights. In particular, we consider the class \cn={WtρWt1ρρW1:W1,,Wt1Md×d,WtM1,d} where the spectral norm of each Wi is bounded by O(1), the Frobenius norm is bounded by R, and ρ is the sigmoid function ex1+ex or the smoothened ReLU function ln(1+ex). We show that for any depth t, if the inputs are in [1,1]d, the sample complexity of \cn is ˜O(dR2ϵ2). This bound is optimal up to log-factors, and substantially improves over the previous state of the art of ˜O(d2R2ϵ2), that was established in a recent line of work. We furthermore show that this bound remains valid if instead of considering the magnitude of the Wi's, we consider the magnitude of WiW0i, where W0i are some reference matrices, with spectral norm of O(1). By taking the W0i to be the matrices in the onset of the training process, we get sample complexity bounds that are sub-linear in the number of parameters, in many {\em typical} regimes of parameters. To establish our results we develop a new technique to analyze the sample complexity of families \ch of predictors. We start by defining a new notion of a randomized approximate description of functions f:\cx\realsd. We then show that if there is a way to approximately describe functions in a class \ch using d bits, then dϵ2 examples suffices to guarantee uniform convergence. Namely, that the empirical loss of all the functions in the class is ϵ-close to the true loss. Finally, we develop a set of tools for calculating the approximate description length of classes of functions that can be presented as a composition of linear function classes and non-linear functions.

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