Fisher-Rao Metric, Geometry, and Complexity of Neural Networks (1711.01530v2)

Abstract: We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity --- the Fisher-Rao norm --- that possesses desirable invariance properties and is motivated by Information Geometry. We discover an analytical characterization of the new capacity measure, through which we establish norm-comparison inequalities and further show that the new measure serves as an umbrella for several existing norm-based complexity measures. We discuss upper bounds on the generalization error induced by the proposed measure. Extensive numerical experiments on CIFAR-10 support our theoretical findings. Our theoretical analysis rests on a key structural lemma about partial derivatives of multi-layer rectifier networks.

• The paper introduces the Fisher-Rao norm as a novel capacity measure that unifies and lower-bounds traditional norms while remaining invariant under re-parametrizations.
• It leverages structural lemmas in multi-layer rectifier networks to analytically characterize large-margin properties and generalization behavior.
• Extensive experiments on CIFAR-10 demonstrate that the Fisher-Rao norm reliably predicts improved generalization, challenging existing flatness-based measures.

Fisher-Rao Metric, Geometry, and Complexity of Neural Networks

The paper "Fisher-Rao Metric, Geometry, and Complexity of Neural Networks" by Liang, Poggio, Rakhlin, and Stokes offers an extensive analysis into the interaction of geometry and capacity measures, specifically in the context of deep neural networks. The authors introduce a novel concept of network capacity via the Fisher-Rao norm, rooted in the principles of Information Geometry. This paper ambitiously sets out to bridge existing norm-based complexity measures, proposing the Fisher-Rao norm as a comprehensive framework that encompasses traditional measures like spectral norm, path norm, and group norms, providing a more invariant and stable lens through which to view neural network capacity.

The central contribution of this paper lies in proposing a new capacity measure—the Fisher-Rao norm—that offers desired invariance properties, facilitating a deeper understanding of the empirical success of deep learning frameworks. The authors leverage information geometry to support their definition of the Fisher-Rao norm, aiming to address shortcomings in prior complexity measures and offer improved generalization error bounds in neural networks.

Key Insights

1. Geometric Invariance and Generalization: The paper thoroughly discusses the significance of geometric invariance in assessing generalization. The Fisher-Rao norm developed in this paper exhibits invariance under linear re-parametrizations, a property distinguishing it from conventional measures that fail in invariance under transformations like node-wise rescaling, especially in rectified networks.
2. Analytical Characterization: By employing a structural lemma concerning partial derivatives specific to multi-layer rectifier networks, the authors establish a pivotal identity that aids in defining the Fisher-Rao norm. This structural insight provides a theoretical underpinning for large-margin properties at stationary points and other generalization-related phenomena, facilitating new analytic avenues.
3. Norm Comparison and Capacity Control: Through comparisons among various established norms, it is shown that the Fisher-Rao norm acts as a lower bound for them, thereby framing an umbrella capacity measure that reflects the capacity controlling traits of these diverse norms in a single framework. The inherent geometry of the Fisher-Rao norm, exhibited through norm comparison inequalities, affords rigorous generalization theorem proofs utilizing this capacity measure for deep linear networks.
4. Extensive Numerical Experiments: The paper substantiates theoretical findings through experiments on CIFAR-10, evaluating the Fisher-Rao norm against standard benchmarks and illustrating its robust generalization properties. Particularly, it examines the norm behavior as networks are over-parametrized by width or depth and under scenarios of vain label assignments, reinforcing the argument for the Fisher-Rao norm's reliability and utility.

Implications and Future Directions

The proposed framework's implications suggest a paradigm facilitating reliable predictions concerning over-parametrized deep networks. Moreover, it challenges existing notions of flatness correlating to generalization, suggesting a fundamental rethinking with respect to invariant geometric properties. By advocating the Fisher-Rao norm, the research potentially paves pathways for a consistent approach in network tuning and algorithmic convergence absent in prior methodologies.

Looking forward, this framework invites further validation across varied architectures beyond rectifiers. Additionally, exploring the Fisher-Rao norm's adaptation with alternate loss functions, myriad network types, and broader datasets can unfold its scalability and versatility. This explores intersections with ongoing discussions in learning theory, positioning the Fisher-Rao metric as a versatile building block for future theoretical explorations and practical implementations in deep learning systems.

Overall, this paper contributes a significant advancement in comprehending deep neural network capacities, with the Fisher-Rao norm promising a refined perspective that balances geometric insights with computational pragmatism. The engagement with invariance and norm-comparison exemplifies a meticulous approach deserving of recognition and utility within computational learning environments.