${\mathrm{\ell }}_1$-regularized Neural Networks are Improperly Learnable in Polynomial Time

We study the improper learning of multi-layer neural networks. Suppose that the neural network to be learned has $k$ hidden layers and that the ${\mathrm{\ell }}_1$-norm of the incoming weights of any neuron is bounded by $L$. We present a kernel-based method, such that with probability at least $1 - \delta$,itlearnsapredictorwhosegeneralizationerrorisatmost$\epsilon$ worse than that of the neural network. The sample complexity and the time complexity of the presented method are polynomial in the input dimension and in $(1\mathrm{/}ϵ,\log (1\mathrm{/}\delta ),F(k,L))$, where $F(k,L)$ is a function depending on $(k,L)$ and on the activation function, independent of the number of neurons. The algorithm applies to both sigmoid-like activation functions and ReLU-like activation functions. It implies that any sufficiently sparse neural network is learnable in polynomial time.