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# On the Learnability of Fully-Connected Neural Networks

*Proceedings of the 20th International Conference on Artificial Intelligence and Statistics*, PMLR 54:83-91, 2017.

#### Abstract

Despite the empirical success of deep neural networks, there is limited theoretical understanding on the learnability of these models using a polynomial-time algorithm. In this paper, we characterize the learnability of fully-connected neural networks via both positive and negative results. We focus on $\ell_1$-regularized networks, where the $\ell_1$-norm of the incoming weights of every neuron is assumed to be bounded by a constant $B > 0$. Our first result shows that such networks are properly learnable in $\text{poly}(n,d,\exp(1/ε^2))$ time, where $n$ and $d$ are the sample size and the input dimension, and $ε> 0$ is the gap to optimality. The bound is achieved by repeatedly sampling over a low-dimensional manifold so as to ensure approximate optimality, but avoids the $\exp(d)$ cost of exhaustively searching over the parameter space. We also establish a hardness result showing that the exponential dependence on $1/ε$ is unavoidable unless $\bf RP = \bf NP$. Our second result shows that the exponential dependence on $1/ε$ can be avoided by exploiting the underlying structure of the data distribution. In particular, if the positive and negative examples can be separated with margin $γ> 0$ by an unknown neural network, then the network can be learned in $\text{poly}(n,d,1/ε)$ time. The bound is achieved by an ensemble method which uses the first algorithm as a weak learner. We further show that the separability assumption can be weakened to tolerate noisy labels. Finally, we show that the exponential dependence on $1/γ$ is unimprovable under a certain cryptographic assumption.