Understanding Certified Training with Interval Bound Propagation

TL;DR

This paper investigates why interval bound propagation (IBP) methods outperform more precise bounds in certifiable neural network training, revealing mechanisms, trade-offs, and conditions that influence robustness and accuracy.

Contribution

The study provides a theoretical and empirical analysis of IBP's success, introduces a new metric for bound tightness, and explores conditions for exact bounds and their regularization effects.

Findings

Wider networks improve IBP-based robustness.

High IBP bound tightness correlates with robustness but is not necessary for it.

Theoretical conditions for exact bounds reveal strong regularization effects.

Abstract

As robustness verification methods are becoming more precise, training certifiably robust neural networks is becoming ever more relevant. To this end, certified training methods compute and then optimize an upper bound on the worst-case loss over a robustness specification. Curiously, training methods based on the imprecise interval bound propagation (IBP) consistently outperform those leveraging more precise bounding methods. Still, we lack an understanding of the mechanisms making IBP so successful. In this work, we thoroughly investigate these mechanisms by leveraging a novel metric measuring the tightness of IBP bounds. We first show theoretically that, for deep linear models, tightness decreases with width and depth at initialization, but improves with IBP training, given sufficient network width. We, then, derive sufficient and necessary conditions on weight matrices for IBP bounds to become exact and demonstrate that these impose strong regularization, explaining the empirically observed trade-off between robustness and accuracy in certified training. Our extensive experimental evaluation validates our theoretical predictions for ReLU networks, including that wider networks improve performance, yielding state-of-the-art results. Interestingly, we observe that while all IBP-based training methods lead to high tightness, this is neither sufficient nor necessary to achieve high certifiable robustness. This hints at the existence of new training methods that do not induce the strong regularization required for tight IBP bounds, leading to improved robustness and standard accuracy.