A Universal Law of Robustness via Isoperimetry (2105.12806v4)

Abstract: Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a partial theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is necessary if one wants to interpolate the data smoothly. Namely we show that smooth interpolation requires $d$ times more parameters than mere interpolation, where $d$ is the ambient data dimension. We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry. In the case of two-layers neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li and Nagaraj. We also give an interpretation of our result as an improved generalization bound for model classes consisting of smooth functions.

• The paper demonstrates that scaling parameters as d×n is crucial for smooth interpolation in deep learning models.
• It establishes that isoperimetric properties of data distributions underpin the concentration of measure needed for robustness.
• It reveals a trade-off between model size and Lipschitz constant, highlighting the need for larger parameter spaces to ensure robustness.

A Universal Law of Robustness via Isoperimetry

The paper authored by Sébastien Bubeck and Mark Sellke presents a theoretical exploration of overparametrization in deep learning through the lens of smooth data interpolation. The central thesis is that overparametrization, contrary to being an inefficiency, is a necessary condition to achieve smooth interpolation for models trained on high-dimensional data. This paper proposes a universal law of robustness, showing that for certain data distributions, the number of parameters must scale proportionally with the product of data dimension and number of samples to ensure smoothness and robustness.

Main Contributions

1. Overparametrization Necessity: The authors argue that traditional views on parametrization, which suggest that merely aligning the number of parameters with the number of equations or data points suffices, are insufficient in the context of deep learning. They demonstrate that for smooth interpolation — critical in achieving robust predictions — one needs significantly more parameters, specifically $d \times n$, where $d$ is the ambient dimension of the data and $n$ is the number of data points.
2. Isoperimetric Conditions: The law of robustness is defined under conditions where the covariate distribution adheres to isoperimetric properties. These properties ensure a form of concentration of measure that is pivotal when considering high-dimensional data distributions — a typical scenario in deep learning applications.
3. Tradeoff between Model Size and Lipschitz Constant: The paper details a size-robustness tradeoff. Specifically, it posits that models with fewer parameters than suggested by the universal law are inherently non-robust as measured by the Lipschitz constant of the function. This highlights the tradeoff between the complexity (or size) of a model and its robustness.
4. Empirical Correlation with Real Datasets: While the results are primarily theoretical, they find supportive empirical observations on datasets like MNIST and ImageNet. The paper interprets previous empirical studies which show that robustly accurate models tend to require larger parameter spaces than traditionally non-robust ones.

Theoretical Implications

• Extended Theoretical Framework: The conjecture previously limited to two-layer neural networks and Gaussian distributions is expanded to encompass a broader range of function classes and data distributions. This broad applicability underscores the general relevance of the law across various deep learning frameworks.
• Limitations of Classical Theorems: Bridging techniques from fields like algebraic geometry and analysis — such as the inverse function theorem — the work illustrates these classical constraints' susceptibility to high-dimensional data's complexity, necessitating a refined approach to parametrization.

Practical Implications

• Model Architecture: The findings suggest that model architectures should consider deeper and wider configurations, especially when robust predictions in adversarial settings are desired. Practitioners may need to revise conventional neural network designs emphasizing parameter limitations.
• Future AI Systems Design: The insights guide the development of new AI models that balance parameter count and robustness, potentially leading to more stable and secure AI systems in real-world applications.

Speculative Outlook

The concept that the amount of parameter extensity required is linked to the intrinsic data manifold poses interesting questions for future research. Investigating this universal law's applicability across unsupervised learning settings or generative models could unlock new areas of exploration. Moreover, the theoretical underpinning of the law reframes optimization dynamics, potentially giving rise to new algorithmic strategies focusing on robustness.

In conclusion, the paper offers pivotal insights into why deep learning models necessitate extensive parameterization and sets the stage for further explorations into the intersections of high-dimensional data analysis, deep learning architecture, and robust optimization.