Hardness of Learning Neural Networks with Natural Weights
This addresses the discrepancy between theoretical hardness results and practical success in neural networks, but the results are incremental as they extend existing hardness to more realistic weight assumptions.
The paper tackles the problem of learning neural networks with 'natural' weight distributions, showing that for depth-2 networks, most networks are hard to learn under distributions like normal and uniform, with no efficient algorithm provably successful for most weights and all input distributions.
Neural networks are nowadays highly successful despite strong hardness results. The existing hardness results focus on the network architecture, and assume that the network's weights are arbitrary. A natural approach to settle the discrepancy is to assume that the network's weights are "well-behaved" and posses some generic properties that may allow efficient learning. This approach is supported by the intuition that the weights in real-world networks are not arbitrary, but exhibit some "random-like" properties with respect to some "natural" distributions. We prove negative results in this regard, and show that for depth-$2$ networks, and many "natural" weights distributions such as the normal and the uniform distribution, most networks are hard to learn. Namely, there is no efficient learning algorithm that is provably successful for most weights, and every input distribution. It implies that there is no generic property that holds with high probability in such random networks and allows efficient learning.