Background The COVID-19 pandemic has driven demand for forecasts to guide policy and planning. Previous research has suggested that combining forecasts from multiple models into a single "ensemble" forecast can increase the robustness of forecasts. Here we evaluate the real-time application of an open, collaborative ensemble to forecast deaths attributable to COVID-19 in the U.S. Methods Beginning on April 13, 2020, we collected and combined one- to four-week ahead forecasts of cumulative deaths for U.S. jurisdictions in standardized, probabilistic formats to generate real-time, publicly available ensemble forecasts. We evaluated the point prediction accuracy and calibration of these forecasts compared to reported deaths. Results Analysis of 2,512 ensemble forecasts made April 27 to July 20 with outcomes observed in the weeks ending May 23 through July 25, 2020 revealed precise short-term forecasts, with accuracy deteriorating at longer prediction horizons of up to four weeks. At all prediction horizons, the prediction intervals were well calibrated with 92-96% of observations falling within the rounded 95% prediction intervals. Conclusions This analysis demonstrates that real-time, publicly available ensemble forecasts issued in April-July 2020 provided robust short-term predictions of reported COVID-19 deaths in the United States. With the ongoing need for forecasts of impacts and resource needs for the COVID-19 response, the results underscore the importance of combining multiple probabilistic models and assessing forecast skill at different prediction horizons. Careful development, assessment, and communication of ensemble forecasts can provide reliable insight to public health decision makers.
Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing generalization bounds is that they are based on uniform convergence and are independent of the training algorithm. In this work, we derive an algorithm-dependent generalization error bound for deep ReLU networks, and show that under certain assumptions on the data distribution, gradient descent (GD) with proper random initialization is able to train a sufficiently over-parameterized DNN to achieve arbitrarily small generalization error. Our work sheds light on explaining the good generalization performance of over-parameterized deep neural networks.
We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both gradient descent and stochastic gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under mild assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by (stochastic) gradient descent produces a sequence of iterates that stay inside a small perturbation region centering around the initial weights, in which the empirical loss function of deep ReLU networks enjoys nice local curvature properties that ensure the global convergence of (stochastic) gradient descent. Our theoretical results shed light on understanding the optimization for deep learning, and pave the way for studying the optimization dynamics of training modern deep neural networks.
The progression of Parkinson’s disease (PD) seems to vary according to the disease stage, which greatly influences the management of PD patients. However, the underlying mechanism of progression in PD remains unclear. This study was designed to explore the progressive pattern of iron accumulation at different stages in PD patients. Sixty right-handed PD patients and 40 normal controls were recruited. According to the disease stage, 45 patients with Hoehn-Yahr stage ≤ 2.5 and 15 patients with Hoehn-Yahr stage ≥ 3 were grouped into early-stage PD (EPD) and late-stage PD (LPD) groups, respectively. The iron content in the cardinal subcortical nuclei covering the cerebrum, cerebellum and midbrain was measured using quantitative susceptibility mapping (QSM). The substantia nigra pars compacta (SNc) showed significantly increased QSM values in the EPD patients compared with the controls. In the LPD patients, while the SNc continued to show increased QSM values compared with the controls and EPD patients, the regions showing increased QSM values spread to include the substantia nigra pars reticulata (SNr), red nucleus (RN) and globus pallidus (GP). Our data also indicated that iron deposition in the GP internal segment (GPi) was more significant than in the GP external segment. No other regions showed significant changes in QSM values among the groups. Therefore, we were able to confirm a regionally progressive pattern of iron accumulation in the different stages of PD, indicating that iron deposition in the SNc is affected exclusively in the early stages of the disease while the SNr, RN and GP, and particularly the GPi segment, become involved in advanced stages of the disease. This is a preliminary study providing objective evidence of the iron-related progression in PD.
Significance This paper compares the probabilistic accuracy of short-term forecasts of reported deaths due to COVID-19 during the first year and a half of the pandemic in the United States. Results show high variation in accuracy between and within stand-alone models and more consistent accuracy from an ensemble model that combined forecasts from all eligible models. This demonstrates that an ensemble model provided a reliable and comparatively accurate means of forecasting deaths during the COVID-19 pandemic that exceeded the performance of all of the models that contributed to it. This work strengthens the evidence base for synthesizing multiple models to support public-health action.
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