LOT: Layer-wise Orthogonal Training on Improving $\ell_2$ Certified Robustness

TL;DR

This paper introduces LOT, a layer-wise orthogonal training method for deep neural networks that enhances $\, ext{l}_2$-certified robustness and scales efficiently, with improvements in robustness metrics across various architectures and semi-supervised settings.

Contribution

The paper proposes a novel layer-wise orthogonal training method (LOT) for Lipschitz-constrained neural networks, improving certified robustness and training efficiency.

Findings

LOT outperforms baselines in deterministic $\, ext{l}_2$ certified robustness.

LOT scales effectively to deeper neural networks.

Semi-supervised learning with LOT improves robustness on CIFAR-10.

Abstract

Recent studies show that training deep neural networks (DNNs) with Lipschitz constraints are able to enhance adversarial robustness and other model properties such as stability. In this paper, we propose a layer-wise orthogonal training method (LOT) to effectively train 1-Lipschitz convolution layers via parametrizing an orthogonal matrix with an unconstrained matrix. We then efficiently compute the inverse square root of a convolution kernel by transforming the input domain to the Fourier frequency domain. On the other hand, as existing works show that semi-supervised training helps improve empirical robustness, we aim to bridge the gap and prove that semi-supervised learning also improves the certified robustness of Lipschitz-bounded models. We conduct comprehensive evaluations for LOT under different settings. We show that LOT significantly outperforms baselines regarding deterministic l2 certified robustness, and scales to deeper neural networks. Under the supervised scenario, we improve the state-of-the-art certified robustness for all architectures (e.g. from 59.04% to 63.50% on CIFAR-10 and from 32.57% to 34.59% on CIFAR-100 at radius rho = 36/255 for 40-layer networks). With semi-supervised learning over unlabelled data, we are able to improve state-of-the-art certified robustness on CIFAR-10 at rho = 108/255 from 36.04% to 42.39%. In addition, LOT consistently outperforms baselines on different model architectures with only 1/3 evaluation time.