Along line ofwork has established convergence rates ofthe empirical distribution tothe true distribution in spaces as general as unbounded metric spaces [54, 25, 45]). In the Euclidean setting, this is well understood [14,2,18], although, to the best of our knowledge, minimax lower bounds have been proven only recently [45]; this setting intersects with our work in the caseσd = 1,σg = 0, pd =,matchingourminimaxrateofn 1/D+n 1/2.

One-shot Federated Learning (FL) has recently emerged as a promising approach, which allows the central server to learn a model in a single communication round. Despite the low communication cost, existing one-shot FL methods are mostly impractical or face inherent limitations, \eg a public dataset is required, clients' models are homogeneous, and additional data/model information need to be uploaded. To overcome these issues, we propose a novel two-stage \textbf{D}ata-fre\textbf{E} o\textbf{N}e-\textbf{S}hot federated l\textbf{E}arning (DENSE) framework, which trains the global model by a data generation stage and a model distillation stage. DENSE is a practical one-shot FL method that can be applied in reality due to the following advantages:(1) DENSE requires no additional information compared with other methods (except the model parameters) to be transferred between clients and the server;(2) DENSE does not require any auxiliary dataset for training;(3) DENSE considers model heterogeneity in FL, \ie different clients can have different model architectures.Experiments on a variety of real-world datasets demonstrate the superiority of our method.For example, DENSE outperforms the best baseline method Fed-ADI by 5.08\% on CIFAR10 dataset.

This paper explores the application of Hidden Markov Models (HMM) and Long Short-Term Memory (LSTM) neural networks for economic forecasting, focusing on predicting CPI inflation rates. The study explores a new approach that integrates HMM-derived hidden states and means as additional features for LSTM modeling, aiming to enhance the interpretability and predictive performance of the models. The research begins with data collection and preprocessing, followed by the implementation of the HMM to identify hidden states representing distinct economic conditions. Subsequently, LSTM models are trained using the original and augmented data sets, allowing for comparative analysis and evaluation. The results demonstrate that incorporating HMM-derived data improves the predictive accuracy of LSTM models, particularly in capturing complex temporal patterns and mitigating the impact of volatile economic conditions. Additionally, the paper discusses the implementation of Integrated Gradients for model interpretability and provides insights into the economic dynamics reflected in the forecasting outcomes.