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### Adversarially Robust Generalization Requires More Data

Machine learning models are often susceptible to adversarial perturbations of their inputs. Even small perturbations can cause state-of-the-art classifiers with high "standard" accuracy to produce an incorrect prediction with high confidence. To better understand this phenomenon, we study adversarially robust learning from the viewpoint of generalization. We show that already in a simple natural data model, the sample complexity of robust learning can be significantly larger than that of "standard" learning. This gap is information theoretic and holds irrespective of the training algorithm or the model family.

### Boosting Barely Robust Learners: A New Perspective on Adversarial Robustness

We present an oracle-efficient algorithm for boosting the adversarial robustness of barely robust learners. Barely robust learning algorithms learn predictors that are adversarially robust only on a small fraction $\beta \ll 1$ of the data distribution. Our proposed notion of barely robust learning requires robustness with respect to a "larger" perturbation set; which we show is necessary for strongly robust learning, and that weaker relaxations are not sufficient for strongly robust learning. Our results reveal a qualitative and quantitative equivalence between two seemingly unrelated problems: strongly robust learning and barely robust learning.

### Black-box Certification and Learning under Adversarial Perturbations

We formally study the problem of classification under adversarial perturbations, both from the learner's perspective, and from the viewpoint of a third-party who aims at certifying the robustness of a given black-box classifier. We analyze a PAC-type framework of semi-supervised learning and identify possibility and impossibility results for proper learning of VC-classes in this setting. We further introduce and study a new setting of black-box certification under limited query budget. We analyze this for various classes of predictors and types of perturbation. We also consider the viewpoint of a black-box adversary that aims at finding adversarial examples, showing that the existence of an adversary with polynomial query complexity implies the existence of a robust learner with small sample complexity.

### A Spectral View of Adversarially Robust Features

Given the apparent difficulty of learning models that are robust to adversarial perturbations, we propose tackling the simpler problem of developing adversarially robust features. Specifically, given a dataset and metric of interest, the goal is to return a function (or multiple functions) that 1) is robust to adversarial perturbations, and 2) has significant variation across the datapoints. We establish strong connections between adversarially robust features and a natural spectral property of the geometry of the dataset and metric of interest. This connection can be leveraged to provide both robust features, and a lower bound on the robustness of any function that has significant variance across the dataset. Finally, we provide empirical evidence that the adversarially robust features given by this spectral approach can be fruitfully leveraged to learn a robust (and accurate) model.

### Action Robust Reinforcement Learning and Applications in Continuous Control

A policy is said to be robust if it maximizes the reward while considering a bad, or even adversarial, model. In this work we formalize two new criteria of robustness to action uncertainty. Specifically, we consider two scenarios in which the agent attempts to perform an action $\mathbf{a}$, and (i) with probability $\alpha$, an alternative adversarial action $\bar{\mathbf{a}}$ is taken, or (ii) an adversary adds a perturbation to the selected action in the case of continuous action space. We show that our criteria are related to common forms of uncertainty in robotics domains, such as the occurrence of abrupt forces, and suggest algorithms in the tabular case. Building on the suggested algorithms, we generalize our approach to deep reinforcement learning (DRL) and provide extensive experiments in the various MuJoCo domains. Our experiments show that not only does our approach produce robust policies, but it also improves the performance in the absence of perturbations. This generalization indicates that action-robustness can be thought of as implicit regularization in RL problems.