  Theoretical results show that neural networks can be approximated by Gaussian
processes in the infinite-width limit. However, for fully connected networks,
it has been previously shown that for any fixed network width, $n$, the
Gaussian approximation gets worse as the network depth, $d$, increases. Given
that modern networks are deep, this raises the question of how well modern
architectures, like ResNets, are captured by the infinite-width limit. To
provide a better approximation, we study ReLU ResNets in the
infinite-depth-and-width limit, where both depth and width tend to infinity as
their ratio, $d/n$, remains constant. In contrast to the Gaussian
infinite-width limit, we show theoretically that the network exhibits
log-Gaussian behaviour at initialization in the infinite-depth-and-width limit,
with parameters depending on the ratio $d/n$. Using Monte Carlo simulations, we
demonstrate that even basic properties of standard ResNet architectures are
poorly captured by the Gaussian limit, but remarkably well captured by our
log-Gaussian limit. Moreover, our analysis reveals that ReLU ResNets at
initialization are hypoactivated: fewer than half of the ReLUs are activated.
Additionally, we calculate the interlayer correlations, which have the effect
of exponentially increasing the variance of the network output. Based on our
analysis, we introduce Balanced ResNets, a simple architecture modification,
which eliminates hypoactivation and interlayer correlations and is more
amenable to theoretical analysis.
