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This paper presents a general framework for norm-based capacity control for$L_{p,q}$ weight normalized deep neural networks. We establish the upper boundon the Rademacher complexities of this family. With an $L_{p,q}$ normalizationwhere $q\le p^*$, and $1/p+1/p^{*}=1$, we discuss properties of awidth-independent capacity control, which only depends on depth by a squareroot term. We further analyze the approximation properties of $L_{p,q}$ weightnormalized deep neural networks. In particular, for an $L_{1,\infty}$ weightnormalized network, the approximation error can be controlled by the $L_1$ normof the output layer, and the corresponding generalization error only depends onthe architecture by the square root of the depth.

Understanding Weight Normalized Deep Neural Networks with Rectified Linear Units