Frobenius Problem, Geometry and Number Fields

We introduce the classical Frobenius Problem in Mathematics

An Investigation of Different Determinants of Default Risks: Evidence from Chinese Commercial Banks

Default risk was likely one of the most significant factors in the 2008 global financial crisis, which caused major adverse impacts on global financial markets. Banks were significantly affected during the crisis and play a pivotal role in the functioning of the global economy. Therefore, it is hugely important to investigate why banks default as doing so will help mitigate future crises and maintain the stable growth of the banking sector. Until now, however, far too little attention has been paid to the default risks facing commercial banks in China and it is not clear whether CAMELS variables will affect banks’ decisions to default. The purpose of this investigation is to explore the determinants of default risk for commercial banks in China. Based on this purpose, this study examines 226 commercial banks in China between 2013-2018 using Pooled OLS, fixed-effects model, random-effects model, and generalized method of the moments. In addition, the growth rate of NPLs is employed as a proxy to reflect the default risk of commercial banks. According to a quantitative analysis of CAMELS indicators, ownership, and financing differences, the results of this research show that asset quality of commercial banks is the most significant factor in default risk, and that, in the future, listed banks are less likely to face credit risk than unlisted banks

Towards the Flatter Landscape and Better Generalization in Federated Learning under Client-level Differential Privacy

To defend the inference attacks and mitigate the sensitive information leakages in Federated Learning (FL), client-level Differentially Private FL (DPFL) is the de-facto standard for privacy protection by clipping local updates and adding random noise. However, existing DPFL methods tend to make a sharp loss landscape and have poor weight perturbation robustness, resulting in severe performance degradation. To alleviate these issues, we propose a novel DPFL algorithm named DP-FedSAM, which leverages gradient perturbation to mitigate the negative impact of DP. Specifically, DP-FedSAM integrates Sharpness Aware Minimization (SAM) optimizer to generate local flatness models with improved stability and weight perturbation robustness, which results in the small norm of local updates and robustness to DP noise, thereby improving the performance. To further reduce the magnitude of random noise while achieving better performance, we propose DP-FedSAM-$top_k$ by adopting the local update sparsification technique. From the theoretical perspective, we present the convergence analysis to investigate how our algorithms mitigate the performance degradation induced by DP. Meanwhile, we give rigorous privacy guarantees with R\'enyi DP, the sensitivity analysis of local updates, and generalization analysis. At last, we empirically confirm that our algorithms achieve state-of-the-art (SOTA) performance compared with existing SOTA baselines in DPFL.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:2303.1124

Towards More Suitable Personalization in Federated Learning via Decentralized Partial Model Training

Personalized federated learning (PFL) aims to produce the greatest personalized model for each client to face an insurmountable problem--data heterogeneity in real FL systems. However, almost all existing works have to face large communication burdens and the risk of disruption if the central server fails. Only limited efforts have been used in a decentralized way but still suffers from inferior representation ability due to sharing the full model with its neighbors. Therefore, in this paper, we propose a personalized FL framework with a decentralized partial model training called DFedAlt. It personalizes the "right" components in the modern deep models by alternately updating the shared and personal parameters to train partially personalized models in a peer-to-peer manner. To further promote the shared parameters aggregation process, we propose DFedSalt integrating the local Sharpness Aware Minimization (SAM) optimizer to update the shared parameters. It adds proper perturbation in the direction of the gradient to overcome the shared model inconsistency across clients. Theoretically, we provide convergence analysis of both algorithms in the general non-convex setting for decentralized partial model training in PFL. Our experiments on several real-world data with various data partition settings demonstrate that (i) decentralized training is more suitable for partial personalization, which results in state-of-the-art (SOTA) accuracy compared with the SOTA PFL baselines; (ii) the shared parameters with proper perturbation make partial personalized FL more suitable for decentralized training, where DFedSalt achieves most competitive performance.Comment: 26 page

Bayesian Learning of Gas Transport in Three-Dimensional Fracture Networks

Modeling gas flow through fractures of subsurface rock is a particularly challenging problem because of the heterogeneous nature of the material. High-fidelity simulations using discrete fracture network (DFN) models are one methodology for predicting gas particle breakthrough times at the surface, but are computationally demanding. We propose a Bayesian machine learning method that serves as an efficient surrogate model, or emulator, for these three-dimensional DFN simulations. Our model trains on a small quantity of simulation data and, using a graph/path-based decomposition of the fracture network, rapidly predicts quantiles of the breakthrough time distribution. The approach, based on Gaussian Process Regression (GPR), outputs predictions that are within 20-30% of high-fidelity DFN simulation results. Unlike previously proposed methods, it also provides uncertainty quantification, outputting confidence intervals that are essential given the uncertainty inherent in subsurface modeling. Our trained model runs within a fraction of a second, which is considerably faster than other methods with comparable accuracy and multiple orders of magnitude faster than high-fidelity simulations

Association between serum potassium and Parkinson’s disease in the US (NHANES 2005–2020)

BackgroundEvaluating the correlation between serum potassium and Parkinson’s disease (PD) in US adults.MethodsA cross-sectional study was conducted on 20,495 adults aged 40 years or older using NHANES data from 2005 to 2020. The study utilized one-way logistic regression and multifactorial logistic regression to examine the correlation between serum potassium levels and PD. Additionally, a smoothed curve fitting approach was employed to assess the concentration-response relationship between serum potassium and PD. Stratified analyses were carried out to investigate potential interactions between serum potassium levels and PD with variables such as age, sex, race, marital status, education, BMI, smoking and medical conditions like coronary, stroke, diabetes, hypertension, and hypercholesterolemia.ResultsIn this study, a total of 20,495 participants, comprising 403 PD and 20,092 non-PD individuals, were included. After adjusted for covariates, multivariable logistic regression revealed that high serum potassium level was an independent risk factor for PD (OR:1.86, 95% CI:1.45 ~ 2.39, p < 0.01).The linear association between serum potassium and PD was described using fitted smoothing curves. Age, sex, race, education, marital, BMI, coronary, stroke, diabetes, hypertension and hypercholesterolemia were not significantly correlated with this positive connection, according to subgroup analysis and interaction testing (P for interaction >0.05).ConclusionSerum potassium levels are elevated in patients with Parkinson's disease compared to non-PD patients. Additional prospective studies are required to explore the significance of serum potassium levels in individuals with Parkinson's disease