Improved Generalization Bounds for Robust Learning

We consider a model of robust learning in an adversarial environment. The learner gets uncorrupted training data with access to possible corruptions that may be affected by the adversary during testing. The learner's goal is to build a robust classifier that would be tested on future adversarial examples. We use a zero-sum game between the learner and the adversary as our game theoretic framework. The adversary is limited to $k$ possible corruptions for each input. Our model is closely related to the adversarial examples model of Schmidt et al. (2018); Madry et al. (2017). Our main results consist of generalization bounds for the binary and multi-class classification, as well as the real-valued case (regression). For the binary classification setting, we both tighten the generalization bound of Feige, Mansour, and Schapire (2015), and also are able to handle an infinite hypothesis class $H$. The sample complexity is improved from $O(\frac{1}{\epsilon^4}\log(\frac{|H|}{\delta}))$ to $O(\frac{1}{\epsilon^2}(k\log(k)VC(H)+\log\frac{1}{\delta}))$. Additionally, we extend the algorithm and generalization bound from the binary to the multiclass and real-valued cases. Along the way, we obtain results on fat-shattering dimension and Rademacher complexity of $k$-fold maxima over function classes; these may be of independent interest. For binary classification, the algorithm of Feige et al. (2015) uses a regret minimization algorithm and an ERM oracle as a blackbox; we adapt it for the multi-class and regression settings. The algorithm provides us with near-optimal policies for the players on a given training sample.Comment: Appearing at the 30th International Conference on Algorithmic Learning Theory (ALT 2019

Robust Regression for Safe Exploration in Control

We study the problem of safe learning and exploration in sequential control problems. The goal is to safely collect data samples from an operating environment to learn an optimal controller. A central challenge in this setting is how to quantify uncertainty in order to choose provably-safe actions that allow us to collect useful data and reduce uncertainty, thereby achieving both improved safety and optimality. To address this challenge, we present a deep robust regression model that is trained to directly predict the uncertainty bounds for safe exploration. We then show how to integrate our robust regression approach with model-based control methods by learning a dynamic model with robustness bounds. We derive generalization bounds under domain shifts for learning and connect them with safety and stability bounds in control. We demonstrate empirically that our robust regression approach can outperform conventional Gaussian process (GP) based safe exploration in settings where it is difficult to specify a good GP prior

Robust Regression for Safe Exploration in Control

We study the problem of safe learning and exploration in sequential control problems. The goal is to safely collect data samples from an operating environment to learn an optimal controller. A central challenge in this setting is how to quantify uncertainty in order to choose provably-safe actions that allow us to collect useful data and reduce uncertainty, thereby achieving both improved safety and optimality. To address this challenge, we present a deep robust regression model that is trained to directly predict the uncertainty bounds for safe exploration. We then show how to integrate our robust regression approach with model-based control methods by learning a dynamic model with robustness bounds. We derive generalization bounds under domain shifts for learning and connect them with safety and stability bounds in control. We demonstrate empirically that our robust regression approach can outperform conventional Gaussian process (GP) based safe exploration in settings where it is difficult to specify a good GP prior

Robust Regression for Safe Exploration in Control

We study the problem of safe learning and exploration in sequential control problems. The goal is to safely collect data samples from an operating environment to learn an optimal controller. A central challenge in this setting is how to quantify uncertainty in order to choose provably-safe actions that allow us to collect useful data and reduce uncertainty, thereby achieving both improved safety and optimality. To address this challenge, we present a deep robust regression model that is trained to directly predict the uncertainty bounds for safe exploration. We then show how to integrate our robust regression approach with model-based control methods by learning a dynamic model with robustness bounds. We derive generalization bounds under domain shifts for learning and connect them with safety and stability bounds in control. We demonstrate empirically that our robust regression approach can outperform conventional Gaussian process (GP) based safe exploration in settings where it is difficult to specify a good GP prior

Interval Bound Propagation\unicode{x2013}aided Few\unicode{x002d}shot Learning

Few-shot learning aims to transfer the knowledge acquired from training on a diverse set of tasks, from a given task distribution, to generalize to unseen tasks, from the same distribution, with a limited amount of labeled data. The underlying requirement for effective few-shot generalization is to learn a good representation of the task manifold. One way to encourage this is to preserve local neighborhoods in the feature space learned by the few-shot learner. To this end, we introduce the notion of interval bounds from the provably robust training literature to few-shot learning. The interval bounds are used to characterize neighborhoods around the training tasks. These neighborhoods can then be preserved by minimizing the distance between a task and its respective bounds. We further introduce a novel strategy to artificially form new tasks for training by interpolating between the available tasks and their respective interval bounds, to aid in cases with a scarcity of tasks. We apply our framework to both model-agnostic meta-learning as well as prototype-based metric-learning paradigms. The efficacy of our proposed approach is evident from the improved performance on several datasets from diverse domains in comparison to a sizable number of recent competitors

Hedging Complexity in Generalization via a Parametric Distributionally Robust Optimization Framework

Empirical risk minimization (ERM) and distributionally robust optimization (DRO) are popular approaches for solving stochastic optimization problems that appear in operations management and machine learning. Existing generalization error bounds for these methods depend on either the complexity of the cost function or dimension of the random perturbations. Consequently, the performance of these methods can be poor for high-dimensional problems with complex objective functions. We propose a simple approach in which the distribution of random perturbations is approximated using a parametric family of distributions. This mitigates both sources of complexity; however, it introduces a model misspecification error. We show that this new source of error can be controlled by suitable DRO formulations. Our proposed parametric DRO approach has significantly improved generalization bounds over existing ERM and DRO methods and parametric ERM for a wide variety of settings. Our method is particularly effective under distribution shifts and works broadly in contextual optimization. We also illustrate the superior performance of our approach on both synthetic and real-data portfolio optimization and regression tasks.Comment: Preliminary version appeared in AISTATS 202

Generalization Error in Deep Learning

Deep learning models have lately shown great performance in various fields such as computer vision, speech recognition, speech translation, and natural language processing. However, alongside their state-of-the-art performance, it is still generally unclear what is the source of their generalization ability. Thus, an important question is what makes deep neural networks able to generalize well from the training set to new data. In this article, we provide an overview of the existing theory and bounds for the characterization of the generalization error of deep neural networks, combining both classical and more recent theoretical and empirical results