Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansion (2409.06402v2)

Abstract: Understanding the mechanisms behind neural network optimization is crucial for improving network design and performance. While various optimization techniques have been developed, a comprehensive understanding of the underlying principles that govern these techniques remains elusive. Specifically, the role of symmetry breaking, a fundamental concept in physics, has not been fully explored in neural network optimization. This gap in knowledge limits our ability to design networks that are both efficient and effective. Here, we propose the symmetry breaking hypothesis to elucidate the significance of symmetry breaking in enhancing neural network optimization. We demonstrate that a simple input expansion can significantly improve network performance across various tasks, and we show that this improvement can be attributed to the underlying symmetry breaking mechanism. We further develop a metric to quantify the degree of symmetry breaking in neural networks, providing a practical approach to evaluate and guide network design. Our findings confirm that symmetry breaking is a fundamental principle that underpins various optimization techniques, including dropout, batch normalization, and equivariance. By quantifying the degree of symmetry breaking, our work offers a practical technique for performance enhancement and a metric to guide network design without the need for complete datasets and extensive training processes.

• The paper introduces input dimension expansion, showing that adding constant dimensions leads to faster convergence and higher classification accuracy.
• The paper draws parallels with physics by applying symmetry breaking to reduce degenerate loss states, resulting in smoother training dynamics.
• The paper presents the Replica Distance (RD) Metric to quantify symmetry breaking, offering a practical tool for designing more effective network architectures.

Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansion

The paper "Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansion" presents a compelling thesis on the fundamental role of symmetry breaking in neural network optimization, proposing a novel input dimension expansion technique to elucidate and leverage this phenomenon for performance improvement across various tasks. The authors theoretically and empirically demonstrate that symmetry breaking, a well-known concept in physics, underpins many optimization techniques and suggest a metric to quantify it in neural networks.

Key Findings

The paper is structured around three primary findings: the impact of input dimension expansion on neural network performance, the relevance of symmetry breaking as an optimization principle, and the introduction of a metric to measure the degree of symmetry breaking.

Input Dimension Expansion

The primary empirical contribution of this research is the demonstration that expanding the dimensionality of input data by adding constant values can significantly enhance the performance of neural networks. This improvement is observed across a range of tasks including image classification, Physics-Informed Neural Networks (PINNs) for solving PDEs, image coloring, and sentiment analysis.

Specifically, the authors present detailed experiments involving various convolutional neural network (CNN) architectures trained on multiple datasets such as CIFAR-10, CIFAR-100, ImageNet-R, and ImageNet-100. They show that dimension expansion leads to faster convergence and higher accuracy, with notable gains in classification performance. For example, an expanded ResNet-18 model achieved an accuracy comparable to a larger ResNet-101 on CIFAR-10, highlighting the effectiveness of this technique.

Symmetry Breaking in Neural Networks

The authors draw a compelling parallel between symmetry breaking in physics and neural network optimization. By introducing additional dimensions filled with constant values to the input data, they break the inherent symmetries in the neural network's parameter space. This symmetry breaking reduces the number of degenerate states—configurations of network weights that produce the same loss—resulting in a smoother and more navigable loss landscape.

To illustrate this phenomenon, the authors use a simple neural network example and compare the loss landscapes with and without input dimension expansion. They demonstrate that the introduction of an additional dimension leads to fewer degenerate states and smoother training dynamics, similar to the symmetry breaking observed in the Ising model of statistical physics.

Symmetry Breaking Techniques and Their Measurement

The paper further investigates the effects of existing neural network optimization techniques such as dropout, batch normalization, and equivariance in the context of symmetry breaking. Through comprehensive loss landscape analyses, the authors show that these techniques inherently contribute to symmetry breaking, thereby improving optimization.

Moreover, they introduce a novel metric, termed the Replica Distance (RD) Metric, to quantify the degree of symmetry breaking in neural networks. This metric is based on calculating the Wasserstein distance between different weight configurations of the network, providing a practical tool to evaluate the extent of symmetry breaking. Their experiments reveal that models with higher symmetry breaking, as measured by this metric, tend to perform better, demonstrating the utility of the metric in guiding neural network design and optimization.

Implications and Future Directions

Practical Implications

The findings from this paper offer practical techniques for enhancing neural network performance. Input dimension expansion is a straightforward yet effective method that can be applied universally across various models and tasks. It allows for significant performance gains without the need for excessive increases in model complexity or parameter count.

The proposed RD Metric provides researchers and practitioners with a valuable tool to evaluate and design network architectures more effectively. By leveraging this metric, one can better understand the underlying optimization dynamics and make informed decisions to improve model robustness and generalization.

Theoretical Implications

The introduction of symmetry breaking as a fundamental principle in neural network optimization bridges concepts from physics and machine learning, offering a deeper theoretical understanding of the optimization landscape. This interdisciplinary approach enriches both fields, providing new insights that could lead to the development of more efficient and interpretable AI models.

Future Research Directions

The promising results and insights from this study open several avenues for future research. Extending the analysis to a broader range of datasets and neural network architectures will help validate and refine the proposed techniques and metrics. Additionally, exploring alternative formulations of the symmetry breaking metric could yield more nuanced insights into optimization dynamics.

Another interesting direction is the identification and exploitation of equivariances within data using neural networks themselves. By embedding such symmetries directly into network architectures, one can potentially develop models that are inherently more efficient and effective, particularly in domain-specific applications.

Conclusion

This paper presents a thorough investigation into the role of symmetry breaking in neural network optimization and introduces practical techniques and metrics to leverage this principle. The findings highlight the significance of input dimension expansion and provide robust theoretical and empirical evidence supporting the fundamental role of symmetry breaking in enhancing neural network performance. This research represents a significant step towards a deeper understanding of neural network optimization, paving the way for more efficient and interpretable AI systems.