Probable Domain Generalization via Quantile Risk Minimization

Domain generalization (DG) seeks predictors which perform well on unseen test distributions by leveraging labeled training data from multiple related distributions or domains. To achieve this, the standard formulation optimizes for worst-case performance over the set of all possible domains. However, with worst-case shifts very unlikely in practice, this generally leads to overly-conservative solutions. In fact, a recent study found that no DG algorithm outperformed empirical risk minimization in terms of average performance. In this work, we argue that DG is neither a worst-case problem nor an average-case problem, but rather a probabilistic one. To this end, we propose a probabilistic framework for DG, which we call Probable Domain Generalization, wherein our key idea is that distribution shifts seen during training should inform us of probable shifts at test time. To realize this, we explicitly relate training and test domains as draws from the same underlying meta-distribution, and propose a new optimization problem -- Quantile Risk Minimization (QRM) -- which requires that predictors generalize with high probability. We then prove that QRM: (i) produces predictors that generalize to new domains with a desired probability, given sufficiently many domains and samples; and (ii) recovers the causal predictor as the desired probability of generalization approaches one. In our experiments, we introduce a more holistic quantile-focused evaluation protocol for DG, and show that our algorithms outperform state-of-the-art baselines on real and synthetic data.

Toward Certified Robustness Against Real-World Distribution Shifts

We consider the problem of certifying the robustness of deep neural networks against real-world distribution shifts. To do so, we bridge the gap between hand-crafted specifications and realistic deployment settings by proposing a novel neural-symbolic verification framework, in which we train a generative model to learn perturbations from data and define specifications with respect to the output of the learned model. A unique challenge arising from this setting is that existing verifiers cannot tightly approximate sigmoid activations, which are fundamental to many state-of-the-art generative models. To address this challenge, we propose a general meta-algorithm for handling sigmoid activations which leverages classical notions of counter-example-guided abstraction refinement. The key idea is to "lazily" refine the abstraction of sigmoid functions to exclude spurious counter-examples found in the previous abstraction, thus guaranteeing progress in the verification process while keeping the state-space small. Experiments on the MNIST and CIFAR-10 datasets show that our framework significantly outperforms existing methods on a range of challenging distribution shifts.

Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks

Lipschitz constants of neural networks allow for guarantees of robustness in image classification, safety in controller design, and generalizability beyond the training data. As calculating Lipschitz constants is NP-hard, techniques for estimating Lipschitz constants must navigate the trade-off between scalability and accuracy. In this work, we significantly push the scalability frontier of a semidefinite programming technique known as LipSDP while achieving zero accuracy loss. We first show that LipSDP has chordal sparsity, which allows us to derive a chordally sparse formulation that we call Chordal-LipSDP. The key benefit is that the main computational bottleneck of LipSDP, a large semidefinite constraint, is now decomposed into an equivalent collection of smaller ones: allowing Chordal-LipSDP to outperform LipSDP particularly as the network depth grows. Moreover, our formulation uses a tunable sparsity parameter that enables one to gain tighter estimates without incurring a significant computational cost. We illustrate the scalability of our approach through extensive numerical experiments.

Do Deep Networks Transfer Invariances Across Classes?

To generalize well, classifiers must learn to be invariant to nuisance transformations that do not alter an input's class. Many problems have "class-agnostic" nuisance transformations that apply similarly to all classes, such as lighting and background changes for image classification. Neural networks can learn these invariances given sufficient data, but many real-world datasets are heavily class imbalanced and contain only a few examples for most of the classes. We therefore pose the question: how well do neural networks transfer class-agnostic invariances learned from the large classes to the small ones? Through careful experimentation, we observe that invariance to class-agnostic transformations is still heavily dependent on class size, with the networks being much less invariant on smaller classes. This result holds even when using data balancing techniques, and suggests poor invariance transfer across classes. Our results provide one explanation for why classifiers generalize poorly on unbalanced and long-tailed distributions. Based on this analysis, we show how a generative approach for learning the nuisance transformations can help transfer invariances across classes and improve performance on a set of imbalanced image classification benchmarks. Source code for our experiments is available at https://github.com/AllanYangZhou/generative-invariance-transfer.

Probabilistically Robust Learning: Balancing Average- and Worst-case Performance

Many of the successes of machine learning are based on minimizing an averaged loss function. However, it is well-known that this paradigm suffers from robustness issues that hinder its applicability in safety-critical domains. These issues are often addressed by training against worst-case perturbations of data, a technique known as adversarial training. Although empirically effective, adversarial training can be overly conservative, leading to unfavorable trade-offs between nominal performance and robustness. To this end, in this paper we propose a framework called probabilistic robustness that bridges the gap between the accurate, yet brittle average case and the robust, yet conservative worst case by enforcing robustness to most rather than to all perturbations. From a theoretical point of view, this framework overcomes the trade-offs between the performance and the sample-complexity of worst-case and average-case learning. From a practical point of view, we propose a novel algorithm based on risk-aware optimization that effectively balances average- and worst-case performance at a considerably lower computational cost relative to adversarial training. Our results on MNIST, CIFAR-10, and SVHN illustrate the advantages of this framework on the spectrum from average- to worst-case robustness.

Learning Robust Output Control Barrier Functions from Safe Expert Demonstrations

This paper addresses learning safe control laws from expert demonstrations. We assume that appropriate models of the system dynamics and the output measurement map are available, along with corresponding error bounds. We first propose robust output control barrier functions (ROCBFs) as a means to guarantee safety, as defined through controlled forward invariance of a safe set. We then present an optimization problem to learn ROCBFs from expert demonstrations that exhibit safe system behavior, e.g., data collected from a human operator. Along with the optimization problem, we provide verifiable conditions that guarantee validity of the obtained ROCBF. These conditions are stated in terms of the density of the data and on Lipschitz and boundedness constants of the learned function and the models of the system dynamics and the output measurement map. When the parametrization of the ROCBF is linear, then, under mild assumptions, the optimization problem is convex. We validate our findings in the autonomous driving simulator CARLA and show how to learn safe control laws from RGB camera images.

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.

Model-Based Domain Generalization

We consider the problem of domain generalization, in which a predictor is trained on data drawn from a family of related training domains and tested on a distinct and unseen test domain. While a variety of approaches have been proposed for this setting, it was recently shown that no existing algorithm can consistently outperform empirical risk minimization (ERM) over the training domains. To this end, in this paper we propose a novel approach for the domain generalization problem called Model-Based Domain Generalization. In our approach, we first use unlabeled data from the training domains to learn multi-modal domain transformation models that map data from one training domain to any other domain. Next, we propose a constrained optimization-based formulation for domain generalization which enforces that a trained predictor be invariant to distributional shifts under the underlying domain transformation model. Finally, we propose a novel algorithmic framework for efficiently solving this constrained optimization problem. In our experiments, we show that this approach outperforms both ERM and domain generalization algorithms on numerous well-known, challenging datasets, including WILDS, PACS, and ImageNet. In particular, our algorithms beat the current state-of-the-art methods on the very-recently-proposed WILDS benchmark by up to 20 percentage points.

Closing the Closed-Loop Distribution Shift in Safe Imitation Learning

Commonly used optimization-based control strategies such as model-predictive and control Lyapunov/barrier function based controllers often enjoy provable stability, robustness, and safety properties. However, implementing such approaches requires solving optimization problems online at high-frequencies, which may not be possible on resource-constrained commodity hardware. Furthermore, how to extend the safety guarantees of such approaches to systems that use rich perceptual sensing modalities, such as cameras, remains unclear. In this paper, we address this gap by treating safe optimization-based control strategies as experts in an imitation learning problem, and train a learned policy that can be cheaply evaluated at run-time and that provably satisfies the same safety guarantees as the expert. In particular, we propose Constrained Mixing Iterative Learning (CMILe), a novel on-policy robust imitation learning algorithm that integrates ideas from stochastic mixing iterative learning, constrained policy optimization, and nonlinear robust control. Our approach allows us to control errors introduced by both the learning task of imitating an expert and by the distribution shift inherent to deviating from the original expert policy. The value of using tools from nonlinear robust control to impose stability constraints on learned policies is shown through sample-complexity bounds that are independent of the task time-horizon. We demonstrate the usefulness of CMILe through extensive experiments, including training a provably safe perception-based controller using a state-feedback-based expert.

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.