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 \[ \cn = \left\{W_t\circ\rho\circ W_{t-1}\circ\rho\ldots\circ \rho\circ W_{1} : W_1,\ldots,W_{t-1}\in M_{d\times d}, W_t\in M_{1,d} \right\} \] where the spectral norm of each $W_i$ is bounded by $O(1)$, the Frobenius norm is bounded by $R$, and $\rho$ is the sigmoid function $\frac{e^x}{1 + e^x}$ or the smoothened ReLU function $ \ln\left(1 + e^x\right)$. We show that for any depth $t$, if the inputs are in $[-1,1]^d$, the sample complexity of $\cn$ is $\tilde O\left(\frac{dR^2}{\epsilon^2}\right)$. This bound is optimal up to log-factors, and substantially improves over the previous state of the art of $\tilde O\left(\frac{d^2R^2}{\epsilon^2}\right)$, 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 $W_i$'s, we consider the magnitude of $W_i - W_i^0$, where $W_i^0$ are some reference matrices, with spectral norm of $O(1)$. By taking the $W_i^0$ 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 $f:\cx\to\reals^d$. We then show that if there is a way to approximately describe functions in a class $\ch$ using $d$ bits, then $\frac{d}{\epsilon^2}$ examples suffices to guarantee uniform convergence. Namely, that the empirical loss of all the functions in the class is $\epsilon$-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.