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Big data can easily be contaminated by outliers or contain variables with heavy-tailed distributions, which makes many conventional methods inadequate. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to the sample size, dimension and moments for optimal tradeoff between bias and robustness. Our theoretical framework deals with heavy-tailed distributions with bounded th moment for any . We establish a sharp phase transition for robust estimation of regression parameters in both low and high dimensions: when , the estimator admits a sub-Gaussian-type deviation bound without sub-Gaussian assumptions on the data, while only a slower rate is available in the regime and the transition is smooth and optimal. In addition, we extend the methodology to allow both heavy-tailed predictors and observation noise. Simulation studies lend further support to the theory. In a genetic study of cancer cell lines that exhibit heavy-tailedness, the proposed methods are shown to be more robust and predictive. Supplementary materials for this article are available online.

Many economic and causal parameters depend on nonparametric or high dimensional first steps. We give a general construction of locally robust/orthogonal moment functions for GMM, where first steps have no effect, locally, on average moment functions. Using these orthogonal moments reduces model selection and regularization bias, as is important in many applications, especially for machine learning first steps. Also, associated standard errors are robust to misspecification when there is the same number of moment functions as parameters of interest. We use these orthogonal moments and cross-fitting to construct debiased machine learning estimators of functions of high dimensional conditional quantiles and of dynamic discrete choice parameters with high dimensional state variables. We show that additional first steps needed for the orthogonal moment functions have no effect, globally, on average orthogonal moment functions. We give a general approach to estimating those additional first steps. We characterize double robustness and give a variety of new doubly robust moment functions. We give general and simple regularity conditions for asymptotic theory.

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg (Science 327:1389-1391, 2010) as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness—the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (Science 327:1389-1391, 2010) (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. We find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.

We extend the omitted variable bias framework with a suite of tools for sensitivity analysis in regression models that does not require assumptions on the functional form of the treatment assignment mechanism nor on the distribution of the unobserved confounders, naturally handles multiple confounders, possibly acting non-linearly, exploits expert knowledge to bound sensitivity parameters and can be easily computed by using only standard regression results. In particular, we introduce two novel sensitivity measures suited for routine reporting. The robustness value describes the minimum strength of association that unobserved confounding would need to have, both with the treatment and with the outcome, to change the research conclusions. The partial R² of the treatment with the outcome shows how strongly confounders explaining all the residual outcome variation would have to be associated with the treatment to eliminate the estimated effect. Next, we offer graphical tools for elaborating on problematic confounders, examining the sensitivity of point estimates and t-values, as well as ‘extreme scenarios’. Finally, we describe problems with a common ‘benchmarking’ practice and introduce a novel procedure to bound the strength of confounders formally on the basis of a comparison with observed covariates. We apply these methods to a running example that estimates the effect of exposure to violence on attitudes toward peace.

In quantum mechanics performing a measurement is an invasive process which generally disturbs the system. Due to this phenomenon, there exist incompatible quantum measurements, i.e. measurements that cannot be simultaneously performed on a single copy of the system. It is then natural to ask what the most incompatible quantum measurements are. To answer this question, several measures have been proposed to quantify how incompatible a set of measurements is, however their properties are not well-understood. In this work, we develop a general framework that encompasses all the commonly used measures of incompatibility based on robustness to noise. Moreover, we propose several conditions that a measure of incompatibility should satisfy, and investigate whether the existing measures comply with them. We find that some of the widely used measures do not fulfil these basic requirements. We also show that when looking for the most incompatible pairs of measurements, we obtain different answers depending on the exact measure. For one of the measures, we analytically prove that projective measurements onto two mutually unbiased bases are among the most incompatible pairs in every dimension. However, for some of the remaining measures we find that some peculiar measurements turn out to be even more incompatible.

Neural networks are vulnerable to adversarial perturbations, which motivates training procedures with formal robustness guarantees. In this paper, we study one-hidden-layer ReLU networks from the perspective of Wasserstein distributional robustness. Leveraging the network structure, we derive an upper bound for the intractable robust surrogate in the form of a tractable regularized empirical risk objective whose regularizer is computed through a low-rank optimization problem based on Burer–Monteiro factorization. This reformulation yields a distributional robustness certificate on the worst-case expected loss over a Wasserstein ball. The upper bound construction and distributional certificate are developed for the shallow fixed-output multiclass formulation, while the optimization analysis focuses on a binary specialization with margin loss and exact linear separability. We also analyze a modified stochastic gradient descent scheme for the resulting regularized problem in this binary linearly separable setting, and we establish a corresponding generalization bound. The experiments validate the proposed surrogate and training procedure on binary MNIST and CIFAR-10 tasks, and we added a 10-class MNIST experiment to further check the multiclass trainability of the surrogate.

Geometric properties of wave functions can explain the appearance of topological invariants in many condensed-matter and quantum systems1. For example, topological invariants describe the plateaux observed in the quantized Hall effect and the pumped charge in its dynamic analogue—the Thouless pump2–4. However, the presence of interparticle interactions can affect the topology of a material, invalidating the idealized formulation in terms of Bloch waves. Despite pioneering experiments in different platforms5–9, the study of topological matter under variations in interparticle interactions has proven challenging10. Here we experimentally realize a topological Thouless pump with fully tuneable Hubbard interactions in an optical lattice and observe regimes with robust pumping, as well as an interaction-induced breakdown. We confirm the pump’s robustness against interactions that are smaller than the protecting gap for both repulsive and attractive interactions. Furthermore, we identify that bound pairs of fermions are responsible for quantized transport at strongly attractive interactions. However, for strong repulsive interactions, topological pumping breaks down, but we show how to reinstate it by modifying the pump trajectory. Our results will prove useful for further investigations of interacting topological matter10, including edge effects11 and interaction-induced topological phases12–15.Thouless pumping is the quantization of charge transport through the adiabatic variation of a system’s parameters. The robustness and breakdown of pumping under variations in interparticle interactions have now been shown with ultracold atoms in an optical lattice.

The robustness of complex networks against node failure and malicious attack has been of interest for decades, while most of the research has focused on random attack or hub-targeted attack. In many real-world scenarios, however, attacks are neither random nor hub-targeted, but localized, where a group of neighboring nodes in a network are attacked and fail. In this paper we develop a percolation framework to analytically and numerically study the robustness of complex networks against such localized attack. In particular, we investigate this robustness in Erd s-Rényi networks, random-regular networks, and scale-free networks. Our results provide insight into how to better protect networks, enhance cybersecurity, and facilitate the design of more robust infrastructures.

Contestation of international norms has become the new focus of IR norm research. The optimism of the 1990s that fundamental liberal norms would diffuse globally has remained unfulfilled in recent years—even human rights norms have witnessed strong contestation. Time and again, controversy has erupted regarding international norms such as the ban on torture or the Responsibility to Protect. Meanwhile, we know little about how such controversy affects the robustness of norms—whether it contributes to their weakening or to their strengthening. Existing research offers two competing hypotheses: One branch of norm research often conceptualizes contestation as a sign of norm weakening. By contrast, another branch assigns contestation a normative power of its own, which strengthens norms. It does not specify the limits of such normative power, however. In this article, we argue that contestation per se is a poor predictor of norm robustness. The type of contestation a norm faces matters. Contestation can either (1) address the dimension of application of a norm or (2) examine its validity by questioning the righteousness of the claims a norm makes. The article draws on two illustrative case studies of extensively contested norms, the Responsibility to Protect and the ban on commercial whaling. We argue that widespread contestation of the very validity of a norm is likely to lead to a loss of norm robustness. Applicatory contestation, by contrast, can—under specific circumstances—even strengthen it.

The current study investigates the robustness of deep learning models for accurate medical diagnosis systems with a specific focus on their ability to maintain performance in the presence of adversarial or noisy inputs. We examine factors that may influence model reliability, including model complexity, training data quality, and hyperparameters; we also examine security concerns related to adversarial attacks that aim to deceive models along with privacy attacks that seek to extract sensitive information. Researchers have discussed various defenses to these attacks to enhance model robustness, such as adversarial training and input preprocessing, along with mechanisms like data augmentation and uncertainty estimation. Tools and packages that extend the reliability features of deep learning frameworks such as TensorFlow and PyTorch are also being explored and evaluated. Existing evaluation metrics for robustness are additionally being discussed and evaluated. This paper concludes by discussing limitations in the existing literature and possible future research directions to continue enhancing the status of this research topic, particularly in the medical domain, with the aim of ensuring that AI systems are trustworthy, reliable, and stable.