Sunghee Yun (He/Him)

Co-founder & CTO @ Erudio Bio, Inc., Chief Business Development Officer @ WeStory.ai, Advisor & Evangelist @ CryptoLab, Inc., KFAS-Salzburg Global Leadership Initiative Fellow @ Salzburg Global Seminar, Adjunct Professor @ Sogang University, Advisory Professor @ DGIST, Global Advisory Board Member @ Innovative Future Brain-Inspired Intelligence System Semiconductor of Sogang University, Network Expert @ Gerson Lehrman Group

Abstract

The rapid growth of big data and distributed computing environments has created an urgent need for scalable machine learning algorithms that can operate effectively across multiple agents or devices. This tutorial presents the Alternating Direction Method of Multipliers (ADMM) as a powerful framework for solving convex optimization problems in decentralized machine learning settings. ADMM, originally proposed by Gabay, Mercier, Glowinski, and Marrocco in 1976, provides a robust approach that combines the reliability of the method of multipliers with the decomposability essential for parallel and distributed computation.

The theoretical foundation of ADMM lies in its ability to decompose complex optimization problems into smaller, manageable subproblems that can be solved independently across multiple computational nodes. The method operates on problems of the form

\[\begin{array}{ll} \mbox{minimize} &f(x) + g(z) \\ \mbox{subject to} &Ax + Bz = c \end{array}\]

where the augmented Lagrangian

\[L_\rho(x,z,y) = f(x) + g(z) + y^T(Ax + Bz - c) + (\rho/2)\|Ax + Bz - c\|^2_2\]

enables alternating minimization steps. This formulation maintains convergence guarantees while supporting natural decomposition when objective functions are block-separable, making it particularly suitable for scenarios involving Internet of Things (IoT) environments and distributed sensor networks.

Practical applications of ADMM demonstrate its versatility across various machine learning domains. The tutorial covers key applications including Lasso regression, where ADMM efficiently handles L1-regularized linear regression problems with computational complexity that scales favorably with problem size. For sparse inverse covariance selection, ADMM outperforms traditional methods like COVSEL, solving 1000×1000 problems in 3-10 minutes compared to over 25 minutes for smaller 400×400 problems using alternative approaches. The consensus optimization framework enables distributed learning scenarios where multiple agents collaborate to minimize aggregate objective functions while maintaining local data privacy.

The consensus classification framework exemplifies ADMM’s power in distributed machine learning, particularly for support vector machines and other classification tasks. Through consensus constraints \(x_i - z = 0\), where xi represents local variables and z represents global consensus, the method enables coordination among distributed agents without requiring centralized data aggregation. Experimental results show successful convergence even in challenging scenarios where data is split in the worst possible manner, with different agents containing only positive or negative samples. The visualization of epoch progression demonstrates how individual agent solutions gradually converge to a unified global solution.

ADMM’s computational efficiency and scalability make it particularly attractive for modern machine learning applications. The method’s ability to cache matrix factorizations and exploit warm-start strategies significantly reduces computational overhead in iterative processes. Distributed implementations using MPI demonstrate practical scalability, with systems handling 400,000×8,000 dense matrices across 80 subsystems on 10 machines, achieving complete lasso solutions in approximately 6 seconds. This combination of theoretical rigor, practical efficiency, and natural decomposability positions ADMM as an essential tool for the future of decentralized machine learning in an increasingly connected and distributed computational landscape.