Approximation and Estimation for High-Dimensional Deep Learning Networks
This provides a theoretical explanation for the empirical success of overparameterized deep networks, addressing a foundational problem in machine learning theory.
The paper tackles the theoretical basis for the generalization ability of deep neural networks with many parameters, showing that the statistical risk (mean squared predictive error) is upper bounded by $[(L^3 \\log d)/n]^{1/2}$, allowing input dimension $d$ to be much larger than sample size $n$ under certain conditions, with lower bounds indicating near-optimality.
It has been experimentally observed in recent years that multi-layer artificial neural networks have a surprising ability to generalize, even when trained with far more parameters than observations. Is there a theoretical basis for this? The best available bounds on their metric entropy and associated complexity measures are essentially linear in the number of parameters, which is inadequate to explain this phenomenon. Here we examine the statistical risk (mean squared predictive error) of multi-layer networks with $\ell^1$-type controls on their parameters and with ramp activation functions (also called lower-rectified linear units). In this setting, the risk is shown to be upper bounded by $[(L^3 \log d)/n]^{1/2}$, where $d$ is the input dimension to each layer, $L$ is the number of layers, and $n$ is the sample size. In this way, the input dimension can be much larger than the sample size and the estimator can still be accurate, provided the target function has such $\ell^1$ controls and that the sample size is at least moderately large compared to $L^3\log d$. The heart of the analysis is the development of a sampling strategy that demonstrates the accuracy of a sparse covering of deep ramp networks. Lower bounds show that the identified risk is close to being optimal.