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Training Deep Neural Networks (DNNs) with adversarial examples often results in poor generalization to test-time adversarial data. This paper investigates this issue, known as adversarially robust generalization, through the lens of Rademacher complexity. Building upon the studies by Khim and Loh (2018); Yin et al. (2019), numerous works have been dedicated to this problem, yet achieving a satisfactory bound remains an elusive goal. Existing works on DNNs either apply to a surrogate loss instead of the robust loss or yield bounds that are notably looser compared to their standard counterparts. In the latter case, the bounds have a higher dependency on the width m of the DNNs or the dimension d of the data, with an extra factor of at least O(√m) or O(√d). This paper presents upper bounds for adversarial Rademacher complexity of DNNs that match the best-known upper bounds in standard settings, as established in the work of Bartlett et al. (2017), with the dependency on width and dimension being O(ln(dm)). The central challenge addressed is calculating the covering number of adversarial function classes. We aim to construct a new cover that possesses two properties: 1) compatibility with adversarial examples, and 2) precision comparable to covers used in standard settings. To this end, we introduce a new variant of covering number called the uniform covering number, specifically designed and proven to reconcile these two properties. Consequently, our method effectively bridges the gap between Rademacher complexity in robust and standard generalization. 

Abstract The Ediacaran Period (~635–539 Ma) is marked by the emergence and diversification of complex metazoans linked to ocean redox changes, but the processes and mechanism of the redox evolution in the Ediacaran ocean are intensely debated. Here we use mercury isotope compositions from multiple black shale sections of the Doushantuo Formation in South China to reconstruct Ediacaran oceanic redox conditions. Mercury isotopes show compelling evidence for recurrent and spatially dynamic photic zone euxinia (PZE) on the continental margin of South China during time intervals coincident with previously identified ocean oxygenation events. We suggest that PZE was driven by increased availability of sulfate and nutrients from a transiently oxygenated ocean, but PZE may have also initiated negative feedbacks that inhibited oxygen production by promoting anoxygenic photosynthesis and limiting the habitable space for eukaryotes, hence abating the long-term rise of oxygen and restricting the Ediacaran expansion of macroscopic oxygen-demanding animals. 

Abstract The driving forces, kill and recovery mechanisms for the end-Permian mass extinction (EPME), the largest Phanerozoic biological crisis, are under debate. Sedimentary records of mercury enrichment and mercury isotopes have suggested the impact of volcanism on the EPME, yet the causes of mercury enrichment and isotope variations remain controversial. Here, we model mercury isotope variations across the EPME to quantitatively assess the effects of volcanism, terrestrial erosion and photic zone euxinia (PZE, toxic, sulfide-rich conditions). Our numerical model shows that while large-scale volcanism remains the main driver of widespread mercury enrichment, the negative shifts of Δ¹⁹⁹Hg isotope signature across the EPME cannot be fully explained by volcanism or terrestrial erosion as proposed before, but require additional fractionation by marine mercury photoreduction under enhanced PZE conditions. Thus our model provides further evidence for widespread and prolonged PZE as a key kill mechanism for both the EPME and the impeded recovery afterward. 

Accepted and expected to be in press in 2021