The Loss Surfaces of Neural Networks with General Activation Functions (2004.03959v3)

Abstract: The loss surfaces of deep neural networks have been the subject of several studies, theoretical and experimental, over the last few years. One strand of work considers the complexity, in the sense of local optima, of high dimensional random functions with the aim of informing how local optimisation methods may perform in such complicated settings. Prior work of Choromanska et al (2015) established a direct link between the training loss surfaces of deep multi-layer perceptron networks and spherical multi-spin glass models under some very strong assumptions on the network and its data. In this work, we test the validity of this approach by removing the undesirable restriction to ReLU activation functions. In doing so, we chart a new path through the spin glass complexity calculations using supersymmetric methods in Random Matrix Theory which may prove useful in other contexts. Our results shed new light on both the strengths and the weaknesses of spin glass models in this context.

• The paper extends loss surface analysis to general activation functions beyond the traditional ReLU, broadening theoretical understanding.
• It employs supersymmetric methods from Random Matrix Theory to navigate complex optimization landscapes analogous to spin glass models.
• The study refines the applicability of spin glass models by outlining their strengths and limitations in modeling diverse neural network architectures.

The paper "The Loss Surfaces of Neural Networks with General Activation Functions" extends the paper of neural network loss surfaces beyond the ReLU activation function, a commonly used non-linear function in deep learning models. Traditional investigations into the behavior and optimization of neural networks have emphasized the complexity of their loss surfaces, explicitly considering their vast number of local optima.

Key Contributions

1. Generalization Beyond ReLU: The authors address the limitations of previous models which were primarily restricted to ReLU activations. They explore neural networks with more general activation functions, broadening the applicability of their findings and providing a more comprehensive understanding of neural network behavior.
2. Supersymmetric Methods: Incorporating supersymmetric methods from Random Matrix Theory, the researchers navigate the complexity calculations inherent in the spin glass model framework. This approach offers a novel perspective compared to earlier methods, potentially impacting various other areas that require complex optimization analysis.
3. Spin Glass Model Analysis: By applying these advanced methods, the paper reevaluates the strengths and weaknesses of using spin glass models to understand neural network loss surfaces. Although these models have provided valuable insights, the paper highlights their limitations and formulates a refined view of their applicability to diverse neural network architectures.

Implications

The findings imply that while spin glass models offer robust analytical tools for understanding optimization landscapes, their use in neural networks with arbitrary activation functions must be carefully considered. This research could enhance the efficacy of local optimization techniques by providing more realistic models of neural network loss surfaces, which adapt to any activation function used.

This work serves as a foundation for future studies aimed at extending theoretical models to encompass a wider variety of neural network architectures, potentially leading to more efficient training methodologies and a deeper grasp of how neural networks converge to solutions in high-dimensional spaces.