# On ℓp-norm Robustness of Ensemble Stumps and Trees

arXiv: https://arxiv.org/abs/2008.08755 · HTML: https://mlsanity.com/paper/2008.08755/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2020.

## Built on by (most cited first, 2 of 2)

- [OACAL: Finding Module-consistent Specifications to Secure Systems from Weakened User Obligations](https://mlsanity.com/paper/2108.08282/) (2021)
- [Adversarial Example Detection in Deployed Tree Ensembles](https://mlsanity.com/paper/2206.13083/) (2022)

## Abstract (arXiv)

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 ℓ_∞ norm perturbation. To study robustness with respect to a general ℓp 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 ℓp norm perturbations for ensemble decision stumps and trees. For robustness verification of ensemble stumps, we prove that complete verification is NP-complete for p∈(0, ∞) while polynomial time algorithms exist for p=0 or ∞. For p∈(0, ∞) 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 ℓp norm. We demonstrate the first certified defense method for training ensemble stumps and trees with respect to ℓp norm perturbations, and verify its effectiveness empirically on real datasets.