Stable behaviour of infinitely wide deep neural networks
This work contributes to the theoretical understanding of deep neural networks, particularly for researchers in machine learning theory, but it is incremental as it extends existing results from Gaussian to stable distributions.
The authors tackled the problem of characterizing the infinite wide limit of deep neural networks with stable-distributed weights, showing that it converges to a stable process that generalizes previous Gaussian process limits. They provided explicit recursive formulas for computing the parameters of this limiting process.
We consider fully connected feed-forward deep neural networks (NNs) where weights and biases are independent and identically distributed as symmetric centered stable distributions. Then, we show that the infinite wide limit of the NN, under suitable scaling on the weights, is a stochastic process whose finite-dimensional distributions are multivariate stable distributions. The limiting process is referred to as the stable process, and it generalizes the class of Gaussian processes recently obtained as infinite wide limits of NNs (Matthews at al., 2018b). Parameters of the stable process can be computed via an explicit recursion over the layers of the network. Our result contributes to the theory of fully connected feed-forward deep NNs, and it paves the way to expand recent lines of research that rely on Gaussian infinite wide limits.