Heating up decision boundaries: isocapacitory saturation, adversarial scenarios and generalization bounds

TL;DR

This paper introduces a novel approach to analyze classifier decision boundaries using Brownian motion and heat diffusion, revealing complex geometric features and deriving stronger generalization bounds linked to boundary geometry.

Contribution

It bridges potential theory with machine learning to study decision boundary geometry and introduces heat-diffusion metrics for analyzing adversarial robustness and generalization.

Findings

Decision boundaries become flatter under adversarial defenses.

Heat-diffusion metrics detect complex geometric features invisible to distance-based methods.

Stronger generalization bounds are derived from boundary geometric control.

Abstract

In the present work we study classifiers' decision boundaries via Brownian motion processes in ambient data space and associated probabilistic techniques. Intuitively, our ideas correspond to placing a heat source at the decision boundary and observing how effectively the sample points warm up. We are largely motivated by the search for a soft measure that sheds further light on the decision boundary's geometry. En route, we bridge aspects of potential theory and geometric analysis (Mazya, 2011, Grigoryan-Saloff-Coste, 2002) with active fields of ML research such as adversarial examples and generalization bounds. First, we focus on the geometric behavior of decision boundaries in the light of adversarial attack/defense mechanisms. Experimentally, we observe a certain capacitory trend over different adversarial defense strategies: decision boundaries locally become flatter as measured by isoperimetric inequalities (Ford et al, 2019); however, our more sensitive heat-diffusion metrics extend this analysis and further reveal that some non-trivial geometry invisible to plain distance-based methods is still preserved. Intuitively, we provide evidence that the decision boundaries nevertheless retain many persistent "wiggly and fuzzy" regions on a finer scale. Second, we show how Brownian hitting probabilities translate to soft generalization bounds which are in turn connected to compression and noise stability (Arora et al, 2018), and these bounds are significantly stronger if the decision boundary has controlled geometric features.