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1. A Riemannian Alternating Direction Method of Multipliers

   https://doi.org/10.1287/moor.2023.0068  

   Li, Jiaxiang; Ma, Shiqian; Srivastava, Tejes (December 2024, Mathematics of Operations Research)

   We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function considered in the ambient space. This class of problems finds important applications in machine learning and statistics, such as sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an ϵ-stationary point is analyzed under mild assumptions. Existing ADMMs for solving nonconvex problems either do not allow a nonconvex constraint set or do not allow a nonsmooth objective function. Our algorithm is the first ADMM-type algorithm that minimizes a nonsmooth objective over manifold—a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method. Funding: The research of S. Ma was supported in part by the Office of Naval Research [Grant N00014-24-1-2705]; the National Science Foundation [Grants DMS-2243650, CCF-2308597, CCF-2311275, and ECCS-2326591]; the University of California, Davis Center for Data Science and Artificial Intelligence Research Innovative Data Science Seed Funding Program; and Rice University start-up fund. 

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2. Multi-Attribute Graph Estimation With Sparse-Group Non-Convex Penalties

   https://doi.org/10.1109/ACCESS.2025.3566959  

   Tugnait, Jitendra K (January 2025, IEEE Access)

   We consider the problem of inferring the conditional independence graph (CIG) of high-dimensional Gaussian vectors from multi-attribute data. Most existing methods for graph estimation are based on single-attribute models where one associates a scalar random variable with each node. In multi-attribute graphical models, each node represents a random vector. In this paper we provide a unified theoretical analysis of multi-attribute graph learning using a penalized log-likelihood objective function. We consider both convex (sparse-group lasso) and sparse-group non-convex (log-sum and smoothly clipped absolute deviation (SCAD) penalties) penalty/regularization functions. An alternating direction method of multipliers (ADMM) approach coupled with local linear approximation to non-convex penalties is presented for optimization of the objective function. For non-convex penalties, theoretical analysis establishing local consistency in support recovery, local convexity and precision matrix estimation in high-dimensional settings is provided under two sets of sufficient conditions: with and without some irrepresentability conditions. We illustrate our approaches using both synthetic and real-data numerical examples. In the synthetic data examples the sparse-group log-sum penalized objective function significantly outperformed the lasso penalized as well as SCAD penalized objective functions with F1 -score and Hamming distance as performance metrics. 

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3. Improving the Privacy and Accuracy of ADMM-Based Distributed Algorithms

   Zhang, Xueru; Khalili, Mohammad Mahdi; Liu, Mingyan (January 2018, International Conference on Machine Learning)

   Alternating direction method of multiplier (ADMM) is a popular method used to design distributed versions of a machine learning algorithm, whereby local computations are performed on local data with the output exchanged among neighbors in an iterative fashion. During this iterative process the leakage of data privacy arises. A differentially private ADMM was proposed in prior work (Zhang & Zhu, 2017) where only the privacy loss of a single node during one iteration was bounded, a method that makes it difficult to balance the tradeoff between the utility attained through distributed computation and privacy guarantees when considering the total privacy loss of all nodes over the entire iterative process. We propose a perturbation method for ADMM where the perturbed term is correlated with the penalty parameters; this is shown to improve the utility and privacy simultaneously. The method is based on a modified ADMM where each node independently determines its own penalty parameter in every iteration and decouples it from the dual updating step size. The condition for convergence of the modified ADMM and the lower bound on the convergence rate are also derived. 

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4. Stochastic ADMM Based Distributed Machine Learning with Differential Privacy

   https://doi.org/10.1007/978-3-030-37228-6_13  

   Ding, Jiahao; Errapotu, Sai Mounika; Zhang, Haijun; Gong, Yanmin; Pan, Miao; Han, Zhu (December 2019, Lecture notes of the Institute for Computer Sciences Social Informatics and Telecommunications Engineering)

   While embracing various machine learning techniques to make effective decisions in the big data era, preserving the privacy of sensitive data poses significant challenges. In this paper, we develop a privacy-preserving distributed machine learning algorithm to address this issue. Given the assumption that each data provider owns a dataset with different sample size, our goal is to learn a common classifier over the union of all the local datasets in a distributed way without leaking any sensitive information of the data samples. Such an algorithm needs to jointly consider efficient distributed learning and effective privacy preservation. In the proposed algorithm, we extend stochastic alternating direction method of multipliers (ADMM) in a distributed setting to do distributed learning. For preserving privacy during the iterative process, we combine differential privacy and stochastic ADMM together. In particular, we propose a novel stochastic ADMM based privacy-preserving distributed machine learning (PS-ADMM) algorithm by perturbing the updating gradients, that provide differential privacy guarantee and have a low computational cost. We theoretically demonstrate the convergence rate and utility bound of our proposed PS-ADMM under strongly convex objective. Through our experiments performed on real-world datasets, we show that PS-ADMM outperforms other differentially private ADMM algorithms under the same differential privacy guarantee. 

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5. Rényi Differentially Private ADMM for Non-Smooth Regularized Optimization

   https://doi.org/10.1145/3374664.3375733  

   Chen, Chen; Lee, Jaewoo (March 2020, CODASPY '20: Proceedings of the Tenth ACM Conference on Data and Application Security and Privacy)

   null (Ed.)

   In this paper we consider the problem of minimizing composite objective functions consisting of a convex differentiable loss function plus a non-smooth regularization term, such as $$L_1$$ norm or nuclear norm, under Rényi differential privacy (RDP). To solve the problem, we propose two stochastic alternating direction method of multipliers (ADMM) algorithms: ssADMM based on gradient perturbation and mpADMM based on output perturbation. Both algorithms decompose the original problem into sub-problems that have closed-form solutions. The first algorithm, ssADMM, applies the recent privacy amplification result for RDP to reduce the amount of noise to add. The second algorithm, mpADMM, numerically computes the sensitivity of ADMM variable updates and releases the updated parameter vector at the end of each epoch. We compare the performance of our algorithms with several baseline algorithms on both real and simulated datasets. Experimental results show that, in high privacy regimes (small ε), ssADMM and mpADMM outperform baseline algorithms in terms of classification and feature selection performance, respectively. 

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