Expressive Losses for Verified Robustness via Convex Combinations

TL;DR

This paper introduces the concept of expressive loss functions, formalized as convex combinations of bounds, to improve the trade-off between accuracy and robustness in verified adversarial training, achieving state-of-the-art results.

Contribution

It formalizes the expressivity of loss functions via convex combinations and demonstrates their effectiveness in enhancing robustness-accuracy trade-offs.

Findings

Convex combination losses achieve state-of-the-art robustness.

Expressivity of loss functions is crucial for performance.

Better worst-case loss approximations do not always improve robustness.

Abstract

In order to train networks for verified adversarial robustness, it is common to over-approximate the worst-case loss over perturbation regions, resulting in networks that attain verifiability at the expense of standard performance. As shown in recent work, better trade-offs between accuracy and robustness can be obtained by carefully coupling adversarial training with over-approximations. We hypothesize that the expressivity of a loss function, which we formalize as the ability to span a range of trade-offs between lower and upper bounds to the worst-case loss through a single parameter (the over-approximation coefficient), is key to attaining state-of-the-art performance. To support our hypothesis, we show that trivial expressive losses, obtained via convex combinations between adversarial attacks and IBP bounds, yield state-of-the-art results across a variety of settings in spite of their conceptual simplicity. We provide a detailed analysis of the relationship between the over-approximation coefficient and performance profiles across different expressive losses, showing that, while expressivity is essential, better approximations of the worst-case loss are not necessarily linked to superior robustness-accuracy trade-offs.