Generalization Bounds for Neural Networks via Approximate Description Length

We investigate the sample complexity of networks with bounds on the magnitude of its weights. In particular, we consider the class [ H=leftWtcircρcirc ldotscircρcirc W1 :W1,ldots,Wt-1∈ Md, d, Wt∈ M1,dright ] 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 fracex1+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 H is tilde Oleft(fracdR2ε2right). This bound is optimal up to log-factors, and substantially improves over the previous state of the art of tilde Oleft(fracd2R2ε2right). We furthermore show that this bound remains valid if instead of considering the magnitude of the Wi's, we consider the magnitude of Wi - Wi0, where Wi0 are some reference matrices, with spectral norm of O(1). By taking the Wi0 to be the matrices at the onset of the training process, we get sample complexity bounds that are sub-linear in the number of parameters, in many typical regimes of parameters. To establish our results we develop a new technique to analyze the sample complexity of families H of predictors. We start by defining a new notion of a randomized approximate description of functions f:X→ℝd. We then show that if there is a way to approximately describe functions in a class H using d bits, then d/ε2 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.