Understanding the loss surface of neural networks is essential for the design of models with predictable performance and their success in applications. Experimental results suggest that sufficiently deep and wide neural networks are not negatively impacted by suboptimal local minima. Despite recent progress, the reason for this outcome is not fully understood. Could deep networks have very few, if at all, suboptimal local optima? or could all of them be equally good? We provide a construction to show that suboptimal local minima (i.e. non-global ones), even though degenerate, exist for fully connected neural networks with sigmoid activation functions. The local minima obtained by our proposed construction belong to a connected set of local solutions that can be escaped from via a non-increasing path on the loss curve. For extremely wide neural networks with two hidden layers, we prove that every suboptimal local minimum belongs to such a connected set. This provides a partial explanation for the successful application of deep neural networks. In addition, we also characterize under what conditions the same construction leads to saddle points instead of local minima for deep neural networks.
That suboptimal local minima must become rather degenerate if the neural network becomes sufficiently large was observed for networks with one hidden layer in @cite_5 and @cite_13 . Recently, Nguyen and Hein @cite_15 generalized this result to deeper networks containing an extremely wide hidden layer. Our contribution can be considered as a continuation of this work.
@cite_5 @cite_15 @cite_13
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It was assumed proven that two-layer feedforward neural networks with t-1 hidden nodes, when presented with t input patterns, can not have any suboptimal local minima on the error surface. In this paper, however, we shall give a counterexample to this assumption. This counterexample consists of a region of local minima with nonzero error on the error surface of a neural network with three hidden nodes when presented with four patterns (the XOR problem). We will also show that the original proof is valid only when an unusual definition of local minimum is used. The past few years have seen a dramatic increase in the performance of recognition systems thanks to the introduction of deep networks for representation learning. However, the mathematical reasons for this success remain elusive. A key issue is that the neural network training problem is nonconvex, hence optimization algorithms may not return a global minima. This paper provides sufficient conditions to guarantee that local minima are globally optimal and that a local descent strategy can reach a global minima from any initialization. Our conditions require both the network output and the regularization to be positively homogeneous functions of the network parameters, with the regularization being designed to control the network size. Our results apply to networks with one hidden layer, where size is measured by the number of neurons in the hidden layer, and multiple deep subnetworks connected in parallel, where size is measured by the number of subnetworks. We study the connection between the highly non-convex loss function of a simple model of the fully-connected feed-forward neural network and the Hamiltonian of the spherical spin-glass model under the assumptions of: i) variable independence, ii) redundancy in network parametrization, and iii) uniformity. These assumptions enable us to explain the complexity of the fully decoupled neural network through the prism of the results from random matrix theory. We show that for large-size decoupled networks the lowest critical values of the random loss function form a layered structure and they are located in a well-defined band lower-bounded by the global minimum. The number of local minima outside that band diminishes exponentially with the size of the network. We empirically verify that the mathematical model exhibits similar behavior as the computer simulations, despite the presence of high dependencies in real networks. We conjecture that both simulated annealing and SGD converge to the band of low critical points, and that all critical points found there are local minima of high quality measured by the test error. This emphasizes a major difference between largeand small-size networks where for the latter poor quality local minima have nonzero probability of being recovered. Finally, we prove that recovering the global minimum becomes harder as the network size increases and that it is in practice irrelevant as global minimum often leads to overfitting. While the optimization problem behind deep neural networks is highly non-convex, it is frequently observed in practice that training deep networks seems possible without getting stuck in suboptimal points. It has been argued that this is the case as all local minima are close to being globally optimal. We show that this is (almost) true, in fact almost all local minima are globally optimal, for a fully connected network with squared loss and analytic activation function given that the number of hidden units of one layer of the network is larger than the number of training points and the network structure from this layer on is pyramidal.