Harmonic Machine Learning Models are Robust

TL;DR

The paper addresses the problem of assessing ML robustness in a black-box, label-free setting. It introduces Harmonic Robustness, a metric $\gamma$ based on the mean-value property of harmonic functions, computed as $\gamma(x,r) = \left| f(x) - \frac{1}{V r^n} \int_{B(x,r)} f\,dV \right|$, to quantify how closely a model's prediction adheres to harmonic-like smoothness. The authors demonstrate the method on low-dimensional models (GBDT and MLP) where overfitted models exhibit higher $\gamma$ and more convoluted decision boundaries, and extend to high-dimensional image classifiers (ResNet-50 and ViT) using simplex-based ball approximations and 0.1% sampling, finding ViT generally more accurate and robust with a Gamma Map for visualization. The work suggests practical benefits for online monitoring, model comparison, and explainability, offering a lightweight, model-agnostic tool to gauge data drift and adversarial vulnerability while highlighting avenues for integrating algebraic function constraints into ML quality standards.

Abstract

We introduce Harmonic Robustness, a powerful and intuitive method to test the robustness of any machine-learning model either during training or in black-box real-time inference monitoring without ground-truth labels. It is based on functional deviation from the harmonic mean value property, indicating instability and lack of explainability. We show implementation examples in low-dimensional trees and feedforward NNs, where the method reliably identifies overfitting, as well as in more complex high-dimensional models such as ResNet-50 and Vision Transformer where it efficiently measures adversarial vulnerability across image classes.

Figures (13)

• Figure 1: Examples of harmonic functions that appear in Nature: soap filmsharmonic1, electrostatic potentialsharmonic2, and heat flowsharmonic3.
• Figure 2: The first few n-simplices ...
• Figure 3: Decision functions and gamma contours of the two classifiers: GBDT1 (left), and overfit version GBDT2 (right).
• Figure 4: Average anharmoniticity for the GBDT well-fit and overfit models (left) and MLP models (right) at various choices of the radius parameter $r$. It is significantly greater for the overfit function at any choice of radius $r$ (statistical error is too small to show).
• Figure 5: Decision functions and gamma contours of the two classifiers: NN-1 (left), and overfit version NN-2 (right).