On the Learnability of Fully-Connected Neural Networks
Yuchen Zhang, Jason Lee, Martin Wainwright, Michael I. Jordan
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.
BibTeX
@InProceedings{pmlr-v54-zhang17a,
title = {{On the learnability of fully-connected neural networks}},
author = {Zhang, Yuchen and Lee, Jason and Wainwright, Martin and Jordan, Michael I.},
booktitle = {Proceedings of the 20th International Conference on Artificial Intelligence and Statistics},
pages = {83--91},
year = {2017},
editor = {Singh, Aarti and Zhu, Jerry},
volume = {54},
series = {Proceedings of Machine Learning Research},
month = {20--22 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v54/zhang17a/zhang17a.pdf},
url = {https://proceedings.mlr.press/v54/zhang17a.html},
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.}
}