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convergence rate analysis for Deep Ritz Method

Communications in Computational Physics 2022年第4期31卷 1020-1048页

作者： Chenguang Duan Yuling Jiao Yanming Lai Dingwei Li Xiliang Lu Jerry Zhijian Yang School of Mathematics and Statistics Wuhan UniversityWuhan 430072P.R.China Hubei Key Laboratory of Computational Science Wuhan UniversityWuhan 430072P.R.China

Using deep neural networks to solve PDEs has attracted a lot of attentions ***,why the deep learning method works is falling far behind its empirical *** this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary *** establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation *** addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training ***,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.

关键词： Deep Ritz method deep neural networks convergence rate analysis

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Clustered Federated Learning with Weighted Model Aggregation for Imbalanced Data

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China Communications 2022年第8期19卷 41-56页

作者： Dong Wang Naifu Zhang Meixia Tao Department of Electronic Engineering Shanghai Jiao Tong UniversityShanghai 200240China

As a promising edge learning framework in future 6G networks,federated learning(FL)faces a number of technical challenges due to the heterogeneous network environment and diversified user *** imbalance is one of these challenges that can significantly degrade the learning *** deal with data imbalance issue,this work proposes a new learning framework,called clustered federated learning with weighted model aggregation(weighted CFL).Compared with traditional FL,our weighted CFL adaptively clusters the participating edge devices based on the cosine similarity of their local gradients at each training iteration,and then performs weighted per-cluster model ***,the similarity threshold for clustering is adaptive over iterations in response to the time-varying divergence of local ***,the weights for per-cluster model aggregation are adjusted according to the data balance feature so as to speed up the convergence *** results show that the proposed weighted CFL achieves a faster model convergence rate and greater learning accuracy than benchmark methods under the imbalanced data scenario.

关键词： clustered federated learning data imbalance convergence rate analysis model aggregation

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On the Linear convergence of the Approximate Proximal Splitting Method for Non-smooth Convex Optimization

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Journal of the Operations Research Society of China 2014年第2期2卷 123-141页

作者： Mojtaba Kadkhodaie Maziar Sanjabi Zhi-Quan Luo University of Minnesota MinneapolisMNUSA

Consider the problem of minimizing the sum of two convex functions,one being smooth and the other *** this paper,we introduce a general class of approximate proximal splitting(APS)methods for solving such minimization *** in the APS class include many well-known algorithms such as the proximal splitting method,the block coordinate descent method(BCD),and the approximate gradient projection methods for smooth convex *** establish the linear convergence of APS methods under a local error bound *** the latter is known to hold for compressive sensing and sparse group LASSO problems,our analysis implies the linear convergence of the BCD method for these problems without strong convexity assumption.

关键词： Convex optimization Proximal splitting method Block coordinate descent method convergence rate analysis Local error bound

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