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Accelerated gradient methods are a powerful optimization tool in machine learning and statistics but their development has traditionally been driven by heuristic motivations. Recent research, however, has demonstrated that these methods can be derived as discretizations of dynamical systems, which in turn has provided a basis for more systematic investigations, especially into the structure of those dynamical systems and their structure-preserving discretizations. In this work we introduce dynamical systems defined through a contact geometry which are not only naturally suited to the optimization goal but also subsume all previous methods based on geometric dynamical systems. These contact dynamical systems also admit a natural, robust discretization through geometric contact integrators. We demonstrate these features in paradigmatic examples which show that we can indeed obtain optimization algorithms that achieve oracle lower bounds on convergence rates while also improving on previous proposals in terms of stability.

IntroductionOver a third of the communities (39%) in the Central Valley of California, a richly diverse and important agricultural region, are classified as disadvantaged—with inadequate access to healthcare, lower socio-economic status, and higher exposure to air and water pollution. The majority of racial and ethnic minorities are also at higher risk of COVID-19 infection, hospitalization, and death according to the Centers for Disease Control and Prevention. Healthy Central Valley Together established a wastewater-based disease surveillance (WDS) program that aims to achieve greater health equity in the region through partnership with Central Valley communities and the Sewer Coronavirus Alert Network. WDS offers a cost-effective strategy to monitor trends in SARS-CoV-2 community infection rates.MethodsIn this study, we evaluated correlations between public health and wastewater data (represented as SARS-CoV-2 target gene copies normalized by pepper mild mottle virus target gene copies) collected for three Central Valley communities over two periods of COVID-19 infection waves between October 2021 and September 2022. Public health data included clinical case counts at county and sewershed scales as well as COVID-19 hospitalization and intensive care unit admissions. Lag-adjusted hospitalization:wastewater ratios were also evaluated as a retrospective metric of disease severity and corollary to hospitalization:case ratios.ResultsConsistent with other studies, strong correlations were found between wastewater and public health data. However, a significant reduction in case:wastewater ratios was observed for all three communities from the first to the second wave of infections, decreasing from an average of 4.7 ± 1.4 over the first infection wave to 0.8 ± 0.4 over the second.DiscussionThe decline in case:wastewater ratios was likely due to reduced clinical testing availability and test seeking behavior, highlighting how WDS can fill data gaps associated with under-reporting of cases. Overall, the hospitalization:wastewater ratios remained more stable through the two waves of infections, averaging 0.5 ± 0.3 and 0.3 ± 0.4 over the first and second waves, respectively.