Collapse of Deep and Narrow Neural Nets
This addresses a critical training instability for deep and narrow networks in scientific computing, though it is incremental as it builds on known issues like vanishing gradients.
The paper demonstrates that deep and narrow neural networks, even with ReLU activations, tend to converge to erroneous mean or median states of the target function with high probability, particularly in solving partial differential equations with high-order derivatives, and provides a safe region diagram to avoid this collapse.
Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically resolved by the rectified linear unit (ReLU) activation. However, here we show that even for such activation, deep and narrow neural networks (NNs) will converge to erroneous mean or median states of the target function depending on the loss with high probability. Deep and narrow NNs are encountered in solving partial differential equations with high-order derivatives. We demonstrate this collapse of such NNs both numerically and theoretically, and provide estimates of the probability of collapse. We also construct a diagram of a safe region for designing NNs that avoid the collapse to erroneous states. Finally, we examine different ways of initialization and normalization that may avoid the collapse problem. Asymmetric initializations may reduce the probability of collapse but do not totally eliminate it.