In Part I of this work, we have proposed a general framework of decentralized stochastic quasi-Newton methods, which converge linearly to the optimal solution under the assumption that the local Hessian inverse approximations have bounded positive eigenvalues. In Part II, we specify two fully decentralized stochastic quasi-Newton methods, damped regularized limited-memory DFP (Davidon-Fletcher-Powell) and damped limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno), to locally construct such Hessian inverse approximations without extra sampling or communication. Both of the methods use a fixed moving window of $M$ past local gradient approximations and local decision variables to adaptively construct positive definite Hessian inverse approximations with bounded eigenvalues, satisfying the assumption in Part I for the linear convergence. For the proposed damped regularized limited-memory DFP, a regularization term is added to improve the performance. For the proposed damped limited-memory BFGS, a two-loop recursion is applied, leading to low storage and computation complexity. Numerical experiments demonstrate that the proposed quasi-Newton methods are much faster than the existing decentralized stochastic first-order algorithms.
In this paper, a class of Decentralized Approximate Newton (DEAN) methods for addressing convex optimization on a networked system are developed, where nodes in the networked system seek for a consensus that minimizes the sum of their individual objective functions through local interactions only. The proposed DEAN algorithms allow each node to repeatedly perform a local approximate Newton update, which leverages tracking the global Newton direction and dissipating the discrepancies among the nodes. Under less restrictive problem assumptions in comparison with most existing second-order method…
This paper studies a policy optimization problem arising from collaborative multi-agent reinforcement learning in a decentralized setting where agents communicate with their neighbors over an undirected graph to maximize the sum of their cumulative rewards. A novel decentralized natural policy gradient method, dubbed Momentum-based Decentralized Natural Policy Gradient (MDNPG), is proposed, which incorporates natural gradient, momentum-based variance reduction, and gradient tracking into the decentralized stochastic gradient ascent framework. The $\mathcal{O}(n^{-1}ε^{-3})$ sample complexity f…
Decentralized Inexact Cubic Newton Method with Consensus Procedure
Mohamed bin Zayed University of Artificial Intelligence (MBZUAI)（穆罕默德·本·扎耶德人工智能大学）；Moscow Independent Research Institute of Artificial Intelligence（莫斯科独立人工智能研究院）；Trusted AI Research Center, RAS（俄罗斯科学院可信人工智能研究中心）；Innopolis University（因诺波利斯大学）
In this paper, we consider a strongly convex finite-sum minimization problem over a decentralized network and propose a communication-efficient decentralized Newton's method for solving it. We first apply dynamic average consensus (DAC) so that each node is able to use a local gradient approximation and a local Hessian approximation to track the global gradient and Hessian, respectively. Second, since exchanging Hessian approximations is far from communication-efficient, we require the nodes to exchange the compressed ones instead and then apply an error compensation mechanism to correct for t…
This paper considers the decentralized consensus optimization problem defined over a network where each node holds a second-order differentiable local objective function. Our goal is to minimize the summation of local objective functions and find the exact optimal solution using only local computation and neighboring communication. We propose a novel Newton tracking algorithm, where each node updates its local variable along a local Newton direction modified with neighboring and historical information. We investigate the connections between the proposed Newton tracking algorithm and several ex…
In this work, we investigate stochastic quasi-Newton methods for minimizing a finite sum of cost functions over a decentralized network. In Part I, we develop a general algorithmic framework that incorporates stochastic quasi-Newton approximations with variance reduction so as to achieve fast convergence. At each time each node constructs a local, inexact quasi-Newton direction that asymptotically approaches the global, exact one. To be specific, (i) A local gradient approximation is constructed by using dynamic average consensus to track the average of variance-reduced local stochastic gradie…
We consider the problem of training machine learning models on distributed data in a decentralized way. For finite-sum problems, fast single-machine algorithms for large datasets rely on stochastic updates combined with variance reduction. Yet, existing decentralized stochastic algorithms either do not obtain the full speedup allowed by stochastic updates, or require oracles that are more expensive than regular gradients. In this work, we introduce a Decentralized stochastic algorithm with Variance Reduction called DVR. DVR only requires computing stochastic gradients of the local functions, a…
Shenzhen Research Institute of Big Data（深圳大数据研究院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Science and Technology of China（中国科学技术大学）；Hong Kong University of Science and Technology（香港科技大学）
This paper considers the decentralized composite optimization problem. We propose a novel decentralized variance-reduction proximal-gradient algorithmic framework, called PMGT-VR, which is based on a combination of several techniques including multi-consensus, gradient tracking, and variance reduction. The proposed framework relies on an imitation of centralized algorithms and we demonstrate that algorithms under this framework achieve convergence rates similar to that of their centralized counterparts. We also describe and analyze two representative algorithms, PMGT-SAGA and PMGT-LSVRG, and c…
Resilient Decentralized Preconditioned Stochastic Proximal Optimization with Gradient Sketching over Time-Varying Networks
Hubei Normal University（湖北师范大学）；City University of Hong Kong（香港城市大学）；University of New South Wales（新南威尔士大学）；Southwest University（西南大学）；Shenzhen University of Advanced Technology（深圳先进技术研究院）
This paper studies decentralized convex-concave minimax optimization problems of the form $\min_x\max_y f(x,y) \triangleq\frac{1}{m}\sum_{i=1}^m f_i(x,y)$, where $m$ is the number of agents and each local function can be written as $f_i(x,y)=\frac{1}{n}\sum_{j=1}^n f_{i,j}(x,y)$. We propose a novel decentralized optimization algorithm, called multi-consensus stochastic variance reduced extragradient, which achieves the best known stochastic first-order oracle (SFO) complexity for this problem. Specifically, each agent requires $\mathcal O((n+κ\sqrt{n})\log(1/\varepsilon))$ SFO calls for strong…
We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over a Riemannian submanifold of the Euclidean space. The proposed algorithm requires a full gradient and Hessian update at the beginning of each epoch while it performs stochastic variance-reduced updates in the iterations within each epoch. The iteration complexity of $O(ε^{-3/2})$ to obtain an $(ε,\sqrtε)$-second-order stationary point, i.e., a point with the Riemannian gradient norm upper bounded by $ε$ and minimum eigenvalue of Riemannian Hessian lower bounded by $-\sqrtε$, is es…
Fudan University（复旦大学）；Tsinghua University（清华大学）；Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）；The Chinese University of Hong Kong（香港中文大学）
In decentralized learning, a network of nodes cooperate to minimize an overall objective function that is usually the finite-sum of their local objectives, and incorporates a non-smooth regularization term for the better generalization ability. Decentralized stochastic proximal gradient (DSPG) method is commonly used to train this type of learning models, while the convergence rate is retarded by the variance of stochastic gradients. In this paper, we propose a novel algorithm, namely DPSVRG, to accelerate the decentralized training by leveraging the variance reduction technique. The basic ide…
This paper considers decentralized stochastic optimization over a network of $n$ nodes, where each node possesses a smooth non-convex local cost function and the goal of the networked nodes is to find an $ε$-accurate first-order stationary point of the sum of the local costs. We focus on an online setting, where each node accesses its local cost only by means of a stochastic first-order oracle that returns a noisy version of the exact gradient. In this context, we propose a novel single-loop decentralized hybrid variance-reduced stochastic gradient method, called GT-HSGD, that outperforms the …
A Variance-Reduced Stochastic Gradient Tracking Algorithm for Decentralized Optimization with Orthogonality Constraints
State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院科学与工程计算国家重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）
Composite and Stochastic Smoothed Decentralized Bregman Proximal-Gradient Methods
The Chinese University of Hong Kong（香港中文大学）；Xiangtan University（湘潭大学）；Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）；Guangzhou Nanfang College（广州南方学院）
This paper describes a novel algorithmic framework to minimize a finite-sum of functions available over a network of nodes. The proposed framework, that we call~\GTVR, is stochastic and decentralized, and thus is particularly suitable for problems where large-scale, potentially private data, cannot be collected or processed at a centralized server. The \GTVR~framework leads to a family of algorithms with two key ingredients: (i) \textit{local variance reduction}, that enables estimating the local batch gradients from arbitrarily drawn samples of local data; and, (ii) \textit{global gradient tr…
We first propose a decentralized proximal stochastic gradient tracking method (DProxSGT) for nonconvex stochastic composite problems, with data heterogeneously distributed on multiple workers in a decentralized connected network. To save communication cost, we then extend DProxSGT to a compressed method by compressing the communicated information. Both methods need only $\mathcal{O}(1)$ samples per worker for each proximal update, which is important to achieve good generalization performance on training deep neural networks. With a smoothness condition on the expected loss function (but not on…
Prox-DBRO-VR: A Unified Analysis on Byzantine-Resilient Decentralized Stochastic Composite Optimization with Variance Reduction and Non-Asymptotic Convergence Rates
Central South University（中南大学）；City University of Hong Kong（香港城市大学）；University of New South Wales（新南威尔士大学）；Southwest University（西南大学）
This paper considers decentralized minimization of $N:=nm$ smooth non-convex cost functions equally divided over a directed network of $n$ nodes. Specifically, we describe a stochastic first-order gradient method, called GT-SARAH, that employs a SARAH-type variance reduction technique and gradient tracking (GT) to address the stochastic and decentralized nature of the problem. We show that GT-SARAH, with appropriate algorithmic parameters, finds an $ε$-accurate first-order stationary point with $O\big(\max\big\{N^{\frac{1}{2}},n(1-λ)^{-2},n^{\frac{2}{3}}m^{\frac{1}{3}}(1-λ)^{-1}\big\}Lε^{-2}\b…
We study stochastic decentralized optimization for the problem of training machine learning models with large-scale distributed data. We extend the widely used EXTRA and DIGing methods with variance reduction (VR), and propose two methods: VR-EXTRA and VR-DIGing. The proposed VR-EXTRA requires the time of $O((κ_s+n)\log\frac{1}ε)$ stochastic gradient evaluations and $O((κ_b+κ_c)\log\frac{1}ε)$ communication rounds to reach precision $ε$, which are the best complexities among the non-accelerated gradient-type methods, where $κ_s$ and $κ_b$ are the stochastic condition number and batch condition…
There is growing interest in large-scale machine learning and optimization over decentralized networks, e.g. in the context of multi-agent learning and federated learning. Due to the imminent need to alleviate the communication burden, the investigation of communication-efficient distributed optimization algorithms - particularly for empirical risk minimization - has flourished in recent years. A large fraction of these algorithms have been developed for the master/slave setting, relying on a central parameter server that can communicate with all agents. This paper focuses on distributed optim…
We consider the decentralized convex optimization problem, where multiple agents must cooperatively minimize a cumulative objective function, with each local function expressible as an empirical average of data-dependent losses. State-of-the-art approaches for decentralized optimization rely on gradient tracking, where consensus is enforced via a doubly stochastic mixing matrix. Construction of such mixing matrices is not straightforward and requires coordination even prior to the start of the optimization algorithm. This paper puts forth a primal-dual framework for decentralized stochastic op…
Stochastic variance reduced optimization methods are known to be globally convergent while they suffer from slow local convergence, especially when moderate or high accuracy is needed. To alleviate this problem, we propose an optimization algorithm -- which we refer to as a hybrid acceleration scheme -- for a class of proximal variance reduced stochastic optimization algorithms. The proposed optimization scheme combines a fast locally convergent algorithm, such as a quasi--Newton method, with a globally convergent variance reduced stochastic algorithm, for instance SAGA or L--SVRG. Our global …
One fundamental problem in decentralized multi-agent optimization is the trade-off between gradient/sampling complexity and communication complexity. We propose new algorithms whose gradient and sampling complexities are graph topology invariant while their communication complexities remain optimal. For convex smooth deterministic problems, we propose a primal dual sliding (PDS) algorithm that computes an $ε$-solution with $O((\tilde{L}/ε)^{1/2})$ gradient and $O((\tilde{L}/ε)^{1/2}+\|\mathcal{A}\|/ε)$ communication complexities, where $\tilde{L}$ is the smoothness parameter of the objective a…
