Stochastic Alternating Direction Method Of Multipliers Research Articles

Adaptive Stochastic ADMM for Decentralized Reinforcement Learning in Edge IoT

Edge computing provides a promising paradigm to support the implementation of Internet of Things (IoT) by offloading tasks to nearby edge nodes. Meanwhile, the increasing network size makes it impractical for centralized data processing due to limited bandwidth, and consequently a decentralized learning scheme is preferable. Reinforcement learning (RL) has been widely investigated and shown to be a promising solution for decision-making and optimal control processes. For RL in a decentralized setup, edge nodes (agents) connected through a communication network aim to work collaboratively to find a policy to optimize the global reward as the sum of local rewards. However, communication costs, scalability, and adaptation in complex environments with heterogeneous agents may significantly limit the performance of decentralized RL. Alternating direction method of multipliers (ADMM) has a structure that allows for decentralized implementation and has shown faster convergence than gradient descent-based methods. Therefore, we propose an adaptive stochastic incremental ADMM (asI-ADMM) algorithm and apply the asI-ADMM to decentralized RL with edge-computing-empowered IoT networks. We provide convergence properties for the proposed algorithms by designing a Lyapunov function and prove that the asI-ADMM has <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(1/k) + \mathcal {O}(1/M)$ </tex-math></inline-formula> convergence rate, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> are the number of iterations and batch samples, respectively. Then, we test our algorithm with two supervised learning problems. For performance evaluation, we simulate two applications in decentralized RL settings with homogeneous and heterogeneous agents. The experimental results show that our proposed algorithms outperform the state of the art in terms of communication costs and scalability and can well adapt to complex IoT environments.

Loopless Variance Reduced Stochastic ADMM for Equality Constrained Problems in IoT Applications

The alternating direction method of multipliers (ADMMs) is an efficient optimization method for solving equality constrained problems in Internet of Things (IoT) applications. Recently, several stochastic variance reduced ADMM algorithms (e.g., SVRG-ADMM) have made exciting progress, such as linear convergence for strongly convex (SC) problems. However, SVRG-ADMM and its variants have an outer loop where the full gradient at the snapshot is computed, and their outer loop contains an inner loop, in which a large number of variance reduced gradients are estimated from random samples. This loopy design makes these methods more complex to analyze and determine the inner loop length, which must be proportional to the condition number to achieve best convergence, and is often set to <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(n)$ </tex-math></inline-formula> as a suboptimal choice, where <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the number of samples. To tackle these issues, we propose an efficient loopless variance reduced stochastic ADMM algorithm, called LVR-SADMM. In our LVR-SADMM, we remove the outer loop and replace it with a biased coin-flip, in which we update the snapshot with a small probability to trigger the full gradient computation. Moreover, we also theoretically analyze the convergence property of LVR-SADMM, which shows that it enjoys a fast linear convergence rate for SC problems. In particular, we also present an accelerated loopless SVRG-ADMM (LAVR-SADMM) method for both SC and non-SC problems. Various experimental results on many real-world data sets verify that the proposed methods can achieve an average speedup of <inline-formula> <tex-math notation="LaTeX">$2\times $ </tex-math></inline-formula> in the SC case and <inline-formula> <tex-math notation="LaTeX">$5\times $ </tex-math></inline-formula> in the non-SC case over their loopy counterparts, respectively.

Laplacian Smoothing Stochastic ADMMs With Differential Privacy Guarantees

Many machine learning tasks such as structured sparse coding and multi-task learning can be converted into an equality constrained optimization problem. The stochastic alternating direction method of multipliers (SADMM) is a popular algorithm to solve such large-scale problems, and has been successfully used in many real-world applications. However, existing SADMMs fail to take into consideration an important issue in their designs, i.e., protecting sensitive information. To address this challenging issue, this paper proposes a novel differential privacy stochastic ADMM framework for solving equality constrained machine learning problems. In particular, to further lift the utility in privacy-preserving equality constrained optimization, a Laplacian smoothing operation is also introduced into our differential privacy ADMM framework, and it can smooth out the Gaussian noise used in the Gaussian mechanism. Then we propose an efficient differentially private variance reduced stochastic ADMM (DP-VRADMM) algorithm with Laplacian smoothing for both strongly convex and general convex objectives. As a by-product, we also present a new differentially private stochastic ADMM algorithm with DP guarantees. In theory, we provide both private guarantees and utility guarantees for the proposed algorithms, which show that Laplacian smoothing can improve the utility bounds of our algorithms. Experimental results on real-world datasets verify our theoretical results and the effectiveness of our algorithms.

Coded Stochastic ADMM for Decentralized Consensus Optimization With Edge Computing

Big data, including applications with high security requirements, are often collected and stored on multiple heterogeneous devices, such as mobile devices, drones, and vehicles. Due to the limitations of communication costs and security requirements, it is of paramount importance to analyze information in a decentralized manner instead of aggregating data to a fusion center. To train large-scale machine learning models, edge/fog computing is often leveraged as an alternative to centralized learning. We consider the problem of learning model parameters in a multiagent system with data locally processed via distributed edge nodes. A class of minibatch stochastic alternating direction method of multipliers (ADMMs) algorithms is explored to develop the distributed learning model. To address two main critical challenges in distributed learning systems, i.e., communication bottleneck and straggler nodes (nodes with slow responses), error-control-coding-based stochastic incremental ADMM is investigated. Given an appropriate minibatch size, we show that the minibatch stochastic ADMM-based method converges in a rate of O(1/√k), where k denotes the number of iterations. Through numerical experiments, it is revealed that the proposed algorithm is communication efficient, rapidly responding, and robust in the presence of straggler nodes compared with state-of-the-art algorithms.

Accelerated Variance Reduced Stochastic ADMM

Recently, many variance reduced stochastic alternating direction method of multipliers (ADMM) methods (e.g. SAG-ADMM, SDCA-ADMM and SVRG-ADMM) have made exciting progress such as linear convergence rates for strongly convex problems. However, the best known convergence rate for general convex problems is O(1/T) as opposed to O(1/T2) of accelerated batch algorithms, where T is the number of iterations. Thus, there still remains a gap in convergence rates between existing stochastic ADMM and batch algorithms. To bridge this gap, we introduce the momentum acceleration trick for batch optimization into the stochastic variance reduced gradient based ADMM (SVRG-ADMM), which leads to an accelerated (ASVRG-ADMM) method. Then we design two different momentum term update rules for strongly convex and general convex cases. We prove that ASVRG-ADMM converges linearly for strongly convex problems. Besides having a low-iteration complexity as existing stochastic ADMM methods, ASVRG-ADMM improves the convergence rate on general convex problems from O(1/T) to O(1/T2). Our experimental results show the effectiveness of ASVRG-ADMM.

Stochastic learning via optimizing the variational inequalities.

A wide variety of learning problems can be posed in the framework of convex optimization. Many efficient algorithms have been developed based on solving the induced optimization problems. However, there exists a gap between the theoretically unbeatable convergence rate and the practically efficient learning speed. In this paper, we use the variational inequality (VI) convergence to describe the learning speed. To this end, we avoid the hard concept of regret in online learning and directly discuss the stochastic learning algorithms. We first cast the regularized learning problem as a VI. Then, we present a stochastic version of alternating direction method of multipliers (ADMMs) to solve the induced VI. We define a new VI-criterion to measure the convergence of stochastic algorithms. While the rate of convergence for any iterative algorithms to solve nonsmooth convex optimization problems cannot be better than O(1/√t), the proposed stochastic ADMM (SADMM) is proved to have an O(1/t) VI-convergence rate for the l1-regularized hinge loss problems without strong convexity and smoothness. The derived VI-convergence results also support the viewpoint that the standard online analysis is too loose to analyze the stochastic setting properly. The experiments demonstrate that SADMM has almost the same performance as the state-of-the-art stochastic learning algorithms but its O(1/t) VI-convergence rate is capable of tightly characterizing the real learning speed.