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# Vanishing Curvature in Randomly Initialized Deep ReLU Networks

*Proceedings of The 25th International Conference on Artificial Intelligence and Statistics*, PMLR 151:7942-7975, 2022.

#### Abstract

Deep ReLU networks are at the basis of many modern neural architectures. Yet, the loss landscape of such networks and its interaction with state-of-the-art optimizers is not fully understood. One of the most crucial aspects is the landscape at random initialization, which often influences convergence speed dramatically. In their seminal works, Xavier & Bengio, 2010 and He et al., 2015 propose an initialization strategy that is supposed to prevent gradients from vanishing. Yet, we identify some shortcomings of their expectation analysis as network depth increases, and show that the proposed initialization can actually fail to deliver stable gradient norms. More precisely, by leveraging an in-depth analysis of the median of the forward pass, we first show that, with high probability, vanishing gradients cannot be circumvented when the network width scales with less than O(depth). Second, we extend this analysis to second-order derivatives and show that random i.i.d. initialization also gives rise to Hessian matrices with eigenspectra that vanish as networks grow in depth. Whenever this happens, optimizers are initialized in a very flat, saddle point-like plateau, which is particularly hard to escape with stochastic gradient descent (SGD) as its escaping time is inversely related to curvature magnitudes. We believe that this observation is crucial for fully understanding (a) the historical difficulties of training deep nets with vanilla SGD and (b) the success of adaptive gradient methods, which naturally adapt to curvature and thus quickly escape flat plateaus.