The loss surface of deep and wide neural networks (1704.08045v2)

Abstract: 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.

• The paper demonstrates that nearly all local minima are globally optimal in deep, wide networks with analytic activation functions.
• It employs advanced mathematical proofs under pyramidal network architectures where a hidden layer exceeds the number of training points.
• The findings bridge theory and practice, informing design strategies for over-parameterized models in modern deep learning.

The Loss Surface of Deep and Wide Neural Networks: An Analytical Exploration

This paper explores the intricacies of loss surfaces in deep and wide neural networks. It investigates why optimization in such networks can be conducted effectively despite their highly non-convex nature. The hypothesis frequently observed in practice, that local minima are essentially equivalent to global minima, is rigorously examined and substantiated under specific theoretical conditions.

Key Results and Claims

The research demonstrates that for fully connected networks with squared loss and analytic activation functions, virtually all local minima are globally optimal. This holds when the network includes a hidden layer with more units than the number of training points, and the subsequent layers follow a pyramidal structure. The paper extends prior work by being applicable to networks of arbitrary depth, generalizing earlier results demonstrated by Yu and others for single hidden layer networks.

Analytical Framework and Methodology

The authors extend the theoretical understanding of deep learning by leveraging advanced mathematical theorems and analytical proofs. For instance, the paper assumes real analytic activation functions and builds upon classic optimization theory. The results are contingent on the oversight that wide constellations pervade certain layers of the network, a situation achievable in realistic analytical bounds. The research accounts for both linearly separable and independent inputs, broadening the scope of practical application.

What sets this work apart is the bridge formed between theory and empirical observation. The conditions such as $n_k \geq N-1$, where $n_k$ is the number of hidden units, are efficiently argued. The implications of these theoretical insights extend to the design choices in real-world applications. Specifically, the pyramidal structure postulated from layer $k+2$ onward resonates with common architectures like convolutional and deep neural networks, often populated with extensive hidden units even in practical scenarios.

Implications and Future Directions

Practically, these findings suggest that optimizing highly over-parameterized networks, common in modern architectures, could inherently place them near globally optimal solutions. This stands to influence how networks are structured in terms of layer width and depth, potentially guiding the architecture search processes toward configurations naturally supporting the stated theoretical assertions.

Theoretically, the results call for a deeper probe into sparsely connected networks like convolutional neural networks, suggesting a sizeable opportunity for research aimed at generalizing these findings beyond fully connected networks. Furthermore, exploring the implications of slightly relaxing the width conditions while maintaining global optimality remains an energizing path for future inquiry.

Conclusion

This paper's detailed investigation into the loss surface of neural networks provides valuable insights, confirming that broad network layers create conditions whereby the seemingly intractable optimization landscapes become more tenable, with most local minima being globally optimal. This synthesis of mathematical rigor and empirical observation positions the paper as a noteworthy contribution to understanding and optimizing deep learning architectures’ training efficacy.