How Can Deep Neural Networks Fail Even With Global Optima? (2407.16872v1)

Abstract: Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions. The ideal optimization process can achieve global optima. However, do global optima always perform well? If not, how bad can it be? In this work, we aim to: 1) extend the expressive power of shallow neural networks to networks of any depth using a simple trick, 2) construct extremely overfitting deep neural networks that, despite having global optima, still fail to perform well on classification and function approximation problems. Different types of activation functions are considered, including ReLU, Parametric ReLU, and Sigmoid functions. Extensive theoretical analysis has been conducted, ranging from one-dimensional models to models of any dimensionality. Numerical results illustrate our theoretical findings.

• The paper demonstrates that attaining global optima does not ensure effective generalization, as shown by examples of overfitting.
• It extends the universal approximation property to deep networks, revealing that depth can approximate complex functions with minimal extra cost.
• The study analyzes various activation functions and high-dimensional inputs, offering insights to refine network design and training strategies.

Overview of "How Can Deep Neural Networks Fail Even With Global Optima?"

The paper "How Can Deep Neural Networks Fail Even With Global Optima?" by Qingguang Guan provides an in-depth analysis of the potential pitfalls in deep neural networks (DNNs), even when global optima are achieved during optimization. The author explores the discrepancy between successful optimization and the actual effectiveness of the model, highlighting the possibility that DNNs can still fail to perform well on classification and function approximation tasks.

Key Contributions

1. Extension of Universal Approximation: The paper introduces a simple technique to extend the universal approximation property from shallow networks to deeper networks. This method highlights the ability of deep networks to approximate complex functions regardless of their depth, with minimal additional computational complexity.
2. Overfitting and Its Implications: A significant portion of the paper is dedicated to constructing examples of overfitting deep neural networks. These networks attain global optima, achieving zero training loss, but exhibit poor generalization. This finding underscores a critical issue in DNNs where they may perform excellently on training data but not extend this performance to unseen data.
3. Theoretical Insights Across Activations: The research investigates various activation functions, including ReLU, Parametric ReLU, and Sigmoid, across models of varying dimensionality. This theoretical exploration is backed by extensive numerical results showcasing the limitations and vulnerabilities of different activation functions.
4. High-Dimensional Extensions: The paper provides a comprehensive look at high-dimensional inputs, illustrating the potential failures in higher dimensions. These examples are particularly relevant in understanding real-world applications where data dimensionality can significantly impact model performance.

Numerical Results and Bold Claims

The research is supported by numerical results that align with the theoretical findings. For instance, numerous models demonstrate zero training loss yet fail to generalize, having an output of zero outside of the training set. This exemplifies the claim that models can achieve global optima without meaningful generalization.

Implications and Future Directions

This paper has significant implications for the design and training of neural networks. Understanding that global optima do not equate to successful model performance is crucial for developing better training regimes and architectures that balance fit and generalization. The research invites practitioners to pay closer attention to training methodologies, particularly in preventing overfitting.

Furthermore, the exploration into extending universal approximation to deep networks suggests potential areas for future research, such as refining architectures and developing novel training methods that mitigate the identified failures.

Conclusion

In conclusion, Guan's work brings to light the nuanced relationship between optimization success and model performance in DNNs. By providing a theoretical framework paired with practical examples, this paper challenges the assumption that achieving global optima is sufficient for optimal model functionality. Future developments in AI will likely benefit from this deeper understanding, paving the way for more robust and generalizable neural networks.