arXiv subjects

Alexander Robey

Publications and source records attributed to Alexander Robey.

45 records · Page 3Linked to original sources

arXiv2022

Provable tradeoffs in adversarially robust classification

It is well known that machine learning methods can be vulnerable to adversarially-chosen perturbations of their inputs. Despite significant progress in the area, foundational open problems remain. In this paper, we address several key questions. We derive exact and approximate Bayes-optimal robust classifiers for the important setting of two- and three-class Gaussian classification problems with arbitrary imbalance, for $\ell_2$ and $\ell_\infty$ adversaries. In contrast to classical Bayes-optimal classifiers, determining the optimal decisions here cannot be made pointwise and new theoretical approaches are needed. We develop and leverage new tools, including recent breakthroughs from probability theory on robust isoperimetry, which, to our knowledge, have not yet been used in the area. Our results reveal fundamental tradeoffs between standard and robust accuracy that grow when data is imbalanced. We also show further results, including an analysis of classification calibration for convex losses in certain models, and finite sample rates for the robust risk.

cs.LG↗

arXiv2021

Model-Based Domain Generalization

Despite remarkable success in a variety of applications, it is well-known that deep learning can fail catastrophically when presented with out-of-distribution data. Toward addressing this challenge, we consider the domain generalization problem, wherein predictors are trained using data drawn from a family of related training domains and then evaluated on a distinct and unseen test domain. We show that under a natural model of data generation and a concomitant invariance condition, the domain generalization problem is equivalent to an infinite-dimensional constrained statistical learning problem; this problem forms the basis of our approach, which we call Model-Based Domain Generalization. Due to the inherent challenges in solving constrained optimization problems in deep learning, we exploit nonconvex duality theory to develop unconstrained relaxations of this statistical problem with tight bounds on the duality gap. Based on this theoretical motivation, we propose a novel domain generalization algorithm with convergence guarantees. In our experiments, we report improvements of up to 30 percentage points over state-of-the-art domain generalization baselines on several benchmarks including ColoredMNIST, Camelyon17-WILDS, FMoW-WILDS, and PACS.

stat.ML↗

arXiv2021

Adversarial Robustness with Semi-Infinite Constrained Learning

Despite strong performance in numerous applications, the fragility of deep learning to input perturbations has raised serious questions about its use in safety-critical domains. While adversarial training can mitigate this issue in practice, state-of-the-art methods are increasingly application-dependent, heuristic in nature, and suffer from fundamental trade-offs between nominal performance and robustness. Moreover, the problem of finding worst-case perturbations is non-convex and underparameterized, both of which engender a non-favorable optimization landscape. Thus, there is a gap between the theory and practice of adversarial training, particularly with respect to when and why adversarial training works. In this paper, we take a constrained learning approach to address these questions and to provide a theoretical foundation for robust learning. In particular, we leverage semi-infinite optimization and non-convex duality theory to show that adversarial training is equivalent to a statistical problem over perturbation distributions, which we characterize completely. Notably, we show that a myriad of previous robust training techniques can be recovered for particular, sub-optimal choices of these distributions. Using these insights, we then propose a hybrid Langevin Monte Carlo approach of which several common algorithms (e.g., PGD) are special cases. Finally, we show that our approach can mitigate the trade-off between nominal and robust performance, yielding state-of-the-art results on MNIST and CIFAR-10. Our code is available at: https://github.com/arobey1/advbench.

stat.ML↗

arXiv2021

Learning Robust Hybrid Control Barrier Functions for Uncertain Systems

The need for robust control laws is especially important in safety-critical applications. We propose robust hybrid control barrier functions as a means to synthesize control laws that ensure robust safety. Based on this notion, we formulate an optimization problem for learning robust hybrid control barrier functions from data. We identify sufficient conditions on the data such that feasibility of the optimization problem ensures correctness of the learned robust hybrid control barrier functions. Our techniques allow us to safely expand the region of attraction of a compass gait walker that is subject to model uncertainty.

eess.SY↗

arXiv2020

Optimal Algorithms for Submodular Maximization with Distributed Constraints

We consider a class of discrete optimization problems that aim to maximize a submodular objective function subject to a distributed partition matroid constraint. More precisely, we consider a networked scenario in which multiple agents choose actions from local strategy sets with the goal of maximizing a submodular objective function defined over the set of all possible actions. Given this distributed setting, we develop Constraint-Distributed Continuous Greedy (CDCG), a message passing algorithm that converges to the tight $(1-1/e)$ approximation factor of the optimum global solution using only local computation and communication. It is known that a sequential greedy algorithm can only achieve a $1/2$ multiplicative approximation of the optimal solution for this class of problems in the distributed setting. Our framework relies on lifting the discrete problem to a continuous domain and developing a consensus algorithm that achieves the tight $(1-1/e)$ approximation guarantee of the global discrete solution once a proper rounding scheme is applied. We also offer empirical results from a multi-agent area coverage problem to show that the proposed method significantly outperforms the state-of-the-art sequential greedy method.

math.OC↗

arXiv2020

Learning Control Barrier Functions from Expert Demonstrations

Inspired by the success of imitation and inverse reinforcement learning in replicating expert behavior through optimal control, we propose a learning based approach to safe controller synthesis based on control barrier functions (CBFs). We consider the setting of a known nonlinear control affine dynamical system and assume that we have access to safe trajectories generated by an expert - a practical example of such a setting would be a kinematic model of a self-driving vehicle with safe trajectories (e.g., trajectories that avoid collisions with obstacles in the environment) generated by a human driver. We then propose and analyze an optimization-based approach to learning a CBF that enjoys provable safety guarantees under suitable Lipschitz smoothness assumptions on the underlying dynamical system. A strength of our approach is that it is agnostic to the parameterization used to represent the CBF, assuming only that the Lipschitz constant of such functions can be efficiently bounded. Furthermore, if the CBF parameterization is convex, then under mild assumptions, so is our learning process. We end with extensive numerical evaluations of our results on both planar and realistic examples, using both random feature and deep neural network parameterizations of the CBF. To the best of our knowledge, these are the first results that learn provably safe control barrier functions from data.

eess.SY↗

arXiv2020

Learning Hybrid Control Barrier Functions from Data

Motivated by the lack of systematic tools to obtain safe control laws for hybrid systems, we propose an optimization-based framework for learning certifiably safe control laws from data. In particular, we assume a setting in which the system dynamics are known and in which data exhibiting safe system behavior is available. We propose hybrid control barrier functions for hybrid systems as a means to synthesize safe control inputs. Based on this notion, we present an optimization-based framework to learn such hybrid control barrier functions from data. Importantly, we identify sufficient conditions on the data such that feasibility of the optimization problem ensures correctness of the learned hybrid control barrier functions, and hence the safety of the system. We illustrate our findings in two simulations studies, including a compass gait walker.

eess.SY↗

arXiv2020

Model-Based Robust Deep Learning: Generalizing to Natural, Out-of-Distribution Data

While deep learning has resulted in major breakthroughs in many application domains, the frameworks commonly used in deep learning remain fragile to artificially-crafted and imperceptible changes in the data. In response to this fragility, adversarial training has emerged as a principled approach for enhancing the robustness of deep learning with respect to norm-bounded perturbations. However, there are other sources of fragility for deep learning that are arguably more common and less thoroughly studied. Indeed, natural variation such as lighting or weather conditions can significantly degrade the accuracy of trained neural networks, proving that such natural variation presents a significant challenge for deep learning. In this paper, we propose a paradigm shift from perturbation-based adversarial robustness toward model-based robust deep learning. Our objective is to provide general training algorithms that can be used to train deep neural networks to be robust against natural variation in data. Critical to our paradigm is first obtaining a model of natural variation which can be used to vary data over a range of natural conditions. Such models may be either known a priori or else learned from data. In the latter case, we show that deep generative models can be used to learn models of natural variation that are consistent with realistic conditions. We then exploit such models in three novel model-based robust training algorithms in order to enhance the robustness of deep learning with respect to the given model. Our extensive experiments show that across a variety of naturally-occurring conditions and across various datasets, deep neural networks trained with our model-based algorithms significantly outperform both standard deep learning algorithms as well as norm-bounded robust deep learning algorithms.

cs.LG↗

arXiv2018

Optimal Physical Preprocessing for Example-Based Super-Resolution

In example-based super-resolution, the function relating low-resolution images to their high-resolution counterparts is learned from a given dataset. This data-driven approach to solving the inverse problem of increasing image resolution has been implemented with deep learning algorithms. In this work, we explore modifying the imaging hardware in order to collect more informative low-resolution images for better ultimate high-resolution image reconstruction. We show that this "physical preprocessing" allows for improved image reconstruction with deep learning in Fourier ptychographic microscopy. Fourier ptychographic microscopy is a technique allowing for both high resolution and high field-of-view at the cost of temporal resolution. In Fourier ptychographic microscopy, variable illumination patterns are used to collect multiple low-resolution images. These low-resolution images are then computationally combined to create an image with resolution exceeding that of any single image from the microscope. We use deep learning to jointly optimize the illumination pattern with the post-processing reconstruction algorithm for a given sample type, allowing for single-shot imaging with both high resolution and high field-of-view. We demonstrate, with simulated data, that the joint optimization yields improved image reconstruction as compared with sole optimization of the post-processing reconstruction algorithm.

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