On ${\ell }_p$-norm Robustness of Ensemble Stumps and Trees

Abstract

Recent papers have demonstrated that ensemble stumps and trees could be vulnerable to small input perturbations, so robustness verification and defense for those models have become an important research problem. However, due to the structure of decision trees, where each node makes decision purely based on one feature value, all the previous works only consider the ${\ell }_{\infty }$ norm perturbation. To study robustness with respect to a general ${\ell }_p$ norm perturbation, one has to consider the correlation between perturbations on different features, which has not been handled by previous algorithms. In this paper, we study the problem of robustness verification and certified defense with respect to general ${\ell }_p$ norm perturbations for ensemble decision stumps and trees. For robustness verification of ensemble stumps, we prove that complete verification is NP-complete for $p\in (0,\infty )$ while polynomial time algorithms exist for $p=0$ or $\infty$. For $p\in (0,\infty )$ we develop an efficient dynamic programming based algorithm for sound verification of ensemble stumps. For ensemble trees, we generalize the previous multi-level robustness verification algorithm to ${\ell }_p$ norm. We demonstrate the first certified defense method for training ensemble stumps and trees with respect to ${\ell }_p$ norm perturbations, and verify its effectiveness empirically on real datasets.

arXiv commentICML 2020

References in corpus (12)

• ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models
• Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples
• Theoretically Principled Trade-off between Robustness and Accuracy
• Provable defenses against adversarial examples via the convex outer adversarial polytope
• Efficient Neural Network Robustness Certification with General Activation Functions
• Efficient Formal Safety Analysis of Neural Networks
• A Dual Approach to Scalable Verification of Deep Networks
• On Adaptive Attacks to Adversarial Example Defenses
• Evasion and Hardening of Tree Ensemble Classifiers
• HopSkipJumpAttack: A Query-Efficient Decision-Based Attack
• Robust Decision Trees Against Adversarial Examples
• Cost-Aware Robust Tree Ensembles for Security Applications