## On Lp-norm Robustness of Ensemble Decision Stumps and Trees

### Yihan Wang · Huan Zhang · Hongge Chen · Duane Boning · Cho-Jui Hsieh

Keywords: [ Adversarial Examples ] [ Safety ] [ Supervised Learning ] [ Boosting / Ensemble Methods ]

[ Abstract ]
Wed 15 Jul 4 p.m. PDT — 4:45 p.m. PDT
Thu 16 Jul 3 a.m. PDT — 3:45 a.m. PDT

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.

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