Random deep neural networks are biased towards simple functions
This addresses the problem of understanding generalization in deep learning for researchers, providing theoretical support for why random networks favor simple functions, though it is incremental as it builds on existing conjectures.
The paper proves that random deep neural networks with ReLU activations produce binary classifiers biased towards simple functions, characterized by high average Hamming distances (at least sqrt(n/(2π log n))) and linear growth in flips needed to change classification with input length n, confirming a prior conjecture.
We prove that the binary classifiers of bit strings generated by random wide deep neural networks with ReLU activation function are biased towards simple functions. The simplicity is captured by the following two properties. For any given input bit string, the average Hamming distance of the closest input bit string with a different classification is at least sqrt(n / (2π log n)), where n is the length of the string. Moreover, if the bits of the initial string are flipped randomly, the average number of flips required to change the classification grows linearly with n. These results are confirmed by numerical experiments on deep neural networks with two hidden layers, and settle the conjecture stating that random deep neural networks are biased towards simple functions. This conjecture was proposed and numerically explored in [Valle Pérez et al., ICLR 2019] to explain the unreasonably good generalization properties of deep learning algorithms. The probability distribution of the functions generated by random deep neural networks is a good choice for the prior probability distribution in the PAC-Bayesian generalization bounds. Our results constitute a fundamental step forward in the characterization of this distribution, therefore contributing to the understanding of the generalization properties of deep learning algorithms.