Linear Convergence Rate Research Articles

A stochastic averaging gradient algorithm with multi‐step communication for distributed optimization

AbstractThis paper studies distributed convex optimization problems over an undirected network where all nodes cooperate to minimize a sum of local objective functions. Each local objective function is further assumed to be an average of several convex instantaneous functions. By incorporating the stochastic averaging gradient into the distributed first‐order primal‐dual method, a stochastic averaging gradient algorithm with multi‐step communication is proposed to solve the optimization problem. For each node, one randomly selected gradient of an instantaneous function is evaluated per iteration, which effectively reduces the computation cost of the algorithm. Based on conditions of strong convexity and Lipschitz continuity of the local instantaneous functions, a linear convergence rate of the proposed algorithm is guaranteed. Numerical simulations on the logistic regression problem demonstrate the performance of the algorithm and correctness of the theoretical results.

Distributed Optimization with Gradient Descent and Quantized Communication

In this paper, we consider the unconstrained distributed optimization problem, in which the exchange of information in the network is captured by a directed graph topology, thus, nodes can only communicate with their neighbors. Additionally, in our problem, the communication channels among the nodes have limited bandwidth. In order to alleviate this limitation, quantized messages should be exchanged among the nodes. For solving this distributed optimization problem, we combine a gradient descent method with a distributed quantized consensus algorithm (which requires the nodes to exchange quantized messages and converges in a finite number of steps). Specifically, at every optimization step, each node (i) performs a gradient descent step (i.e., subtracts the scaled gradient from its current estimate), and (ii) performs a finite-time calculation of the quantized average of every node's estimate in the network. As a consequence, this algorithm approximately mimics the centralized gradient descent algorithm. We show that our algorithm asymptotically converges to a neighborhood of the optimal solution with linear convergence rate. The performance of the proposed algorithm is demonstrated via simple illustrative examples.

Byzantine-Robust Distributed Learning With Compression

Communication between workers and the master node to collect local stochastic gradients is a key bottleneck in a large-scale distributed learning system. Various recent works have proposed to compress the local stochastic gradients to mitigate the communication overhead. However, robustness to malicious attacks is rarely considered in such a setting. In this work, we investigate the problem of Byzantine-robust compressed distributed learning, where the attacks from Byzantine workers can be arbitrarily malicious. We theoretically point out that different to the attacks-free compressed stochastic gradient descent (SGD), its vanilla combination with geometric median-based robust aggregation seriously suffers from the compression noise in the presence of Byzantine attacks. In light of this observation, we propose to reduce the compression noise with gradient difference compression so as to improve the Byzantine-robustness. We also observe the impact of the intrinsic stochastic noise caused by selecting random samples, and adopt the stochastic average gradient algorithm (SAGA) to gradually eliminate the inner variations of regular workers. We theoretically prove that the proposed algorithm reaches a neighborhood of the optimal solution at a linear convergence rate, and the asymptotic learning error is in the same order as that of the state-of-the-art uncompressed method. Finally, numerical experiments demonstrate the effectiveness of the proposed method.

ALGORITHMS FOR THE RICCATI EQUATION WITH A NONSINGULAR M-MATRIX

We consider different algorithms with linear rate of convergence for computing the minimal nonnegative solution of M-matrix algebraic Riccati equation. The performance of the considered algorithms are illustrated on numerical examples.

АЛГОРИТМ ЕКСТРАПОЛЯЦІЇ З МИНУЛОГО ДЛЯ ВАРІАЦІЙНИХ НЕРІВНОСТЕЙ В ГІЛЬБЕРТОВОМУ ПРОСТОРІ

The article considers variational inequalities with operators acting in a Hilbert space. For these problems, variants of the Extrapolation from the Past method have been proposed and studied. A sub-linear efficiency estimate for the gap function is proved. The strong convergence of the Extrapolation from the Past method for variational inequalities with uniformly monotone operators is proved. The linear rate of convergence of the Extrapolation from the Past method for variational inequalities with operators satisfying the generalized strong monotonicity condition is proved. An adaptive version of the algorithm is proposed. Regularized variants of the algorithm are proposed and theorems on their strong convergence are proved.

Local Geometry of Nonconvex Spike Deconvolution From Low-Pass Measurements

Spike deconvolution is the problem of recovering the point sources from their convolution with a known point spread function, which plays a fundamental role in many sensing and imaging applications. In this paper, we investigate the local geometry of recovering the parameters of point sources$\unicode{x2014}$including both amplitudes and locations$\unicode{x2014}$by minimizing a natural nonconvex least-squares loss function measuring the observation residuals. We propose preconditioned variants of gradient descent (GD), where the search direction is scaled via some carefully designed preconditioning matrices. We begin with a simple fixed preconditioner design, which adjusts the learning rates of the locations at a different scale from those of the amplitudes, and show it achieves a linear rate of convergence$\unicode{x2014}$in terms of entrywise errors$\unicode{x2014}$when initialized close to the ground truth, as long as the separation between the true spikes is sufficiently large. However, the convergence rate slows down significantly when the dynamic range of the source amplitudes is large. To bridge this issue, we introduce an adaptive preconditioner design, which compensates for the learning rates of different sources in an iteration-varying manner based on the current estimate. The adaptive design provably leads to an accelerated convergence rate that is independent of the dynamic range, highlighting the benefit of adaptive preconditioning in nonconvex spike deconvolution. Numerical experiments are provided to corroborate the theoretical findings.

A linearly convergent self-adaptive gradient projection algorithm for sparse signal reconstruction in compressive sensing

<abstract><p>For sparse signal reconstruction (SSR) problem in compressive sensing (CS), by the splitting technique, we first transform it into a continuously differentiable convex optimization problem, and then a new self-adaptive gradient projection algorithm is proposed to solve the SSR problem, which has fast solving speed and pinpoint accuracy when the dimension increases. Global convergence of the proposed algorithm is established in detail. Without any assumptions, we establish global $ R- $linear convergence rate of the proposed algorithm, which is a new result for constrained convex (rather than strictly convex) quadratic programming problem. Furthermore, we can also obtain an approximate optimal solution in a finite number of iterations. Some numerical experiments are made on the sparse signal recovery and image restoration to exhibit the efficiency of the proposed algorithm. Compared with the state-of-the-art algorithms in SSR problem, the proposed algorithm is more accurate and efficient.</p></abstract>

Decentralized Inexact Proximal Gradient Method With Network-Independent Stepsizes for Convex Composite Optimization

This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized algorithm for solving convex composite optimization problems over undirected and connected networks. The local loss function in these problems contains both smooth and nonsmooth terms. The proposed algorithm uses uncoordinated network-independent constant stepsizes and only needs to approximately solve a sequence of proximal mappings, which is advantageous for solving decentralized composite optimization problems where the proximal mappings of the nonsmooth loss functions may not have analytical solutions. For the general convex case, we prove an O(1/k) convergence rate of the proposed algorithm, which can be improved to o(1/k) if the proximal mappings are solved exactly. Furthermore, with metric subregularity, we establish a linear convergence rate for the proposed algorithm. Numerical experiments demonstrate the efficiency of the algorithm.

A Distributed Dynamic Event-Triggered Algorithm With Linear Convergence Rate for the Economic Dispatch Problem

With the rapid development of the scale of distributed energy resources, the implementation of distributed algorithms imposes ever-increasing requirements on communication resources. With the aim of reducing the communication burden required to solve the economic dispatch problem, we here consider three key aspects, namely the amount of information exchanged per broadcast, the broadcast frequency per iteration, and the number of iterations needed to achieve a certain accuracy. The proposed primal-dual based algorithm is integrated with a discrete dynamic event-triggered scheme and enjoys significant advantages in all three mentioned aspects. We prove that the proposed algorithm converges to the optimal point at a linear convergence rate for suitable operating parameters and for cost functions that are strongly convex and smooth. We confirm the effectiveness and demonstrate the advantages of our approach by means of a set of simulation experiments.

A Distributed Nesterov-Like Gradient Tracking Algorithm for Composite Constrained Optimization

This paper focuses on the constrained optimization problem where the objective function is composed of smooth (possibly nonconvex) and nonsmooth parts. The proposed algorithm integrates the successive convex approximation (SCA) technique with the gradient tracking mechanism that aims at achieving a linear convergence rate and employing the momentum term to regulate update directions in each time instant. It is proved that the proposed algorithm converges provided that the constant step size and momentum parameter are lower than the given upper bounds. When the smooth part is strongly convex, the proposed algorithm linearly converges to the global optimal solution, whereas it converges to a local stationary solution with a sub-linear convergence rate if the smooth part is nonconvex. Numerical simulations are applied to demonstrate the validity of the proposed algorithm and the theoretical analysis.

On the convergence analysis of asynchronous SGD for solving consistent linear systems

Parallel and distributed stochastic algorithms have drawn significant attention in the realm of big data and machine learning. While the synchronous versions of these algorithms are well understood in terms of their convergence, the convergence analyses of their asynchronous counterparts are not widely studied. In this paper, we propose and analyze a distributed, asynchronous parallel algorithm to solve an arbitrary, consistent linear system by reformulating the system into a stochastic optimization problem as Richtárik and Takác̃ in [1]. We compare the convergence rates of our asynchronous algorithm with the synchronous parallel algorithm proposed by Richtárik and Takáč in [1] under different choices of the hyperparameters—the stepsize, the damping factor, the number of processors, and the delay factor. We show that our asynchronous parallel algorithm enjoys a global linear convergence rate, similar to the synchronous parallel algorithm in [1] under the same setup. We also show that our asynchronous algorithm improves upon the synchronous algorithm in [1] with a better convergence rate when the number of processors is larger than four. Furthermore, our asynchronous parallel algorithm performs asymptotically better than its synchronous counterpart for certain linear systems. Finally, we compute the minimum number of processors required for those systems for which our asynchronous algorithm is better and find that this number can be as low as two for some ill-conditioned problems.

Convergence rates for the joint solution of inverse problems with compressed sensing data

Compressed sensing (CS) is a powerful tool for reducing the amount of data to be collected while maintaining high spatial resolution. Such techniques work well in practice and at the same time are supported by solid theory. Standard CS results assume measurements to be made directly on the targeted signal. In many practical applications, however, CS information can only be taken from indirect data related to the original signal by an additional forward operator. If inverting the forward operator is ill-posed, then existing CS theory is not applicable. In this paper, we address this issue and present two joint reconstruction approaches, namely relaxed co-regularization and strict co-regularization, for CS from indirect data. As main results, we derive error estimates for recovering and . In particular, we derive a linear convergence rate in the norm for the latter. To obtain these results, solutions are required to satisfy a source condition and the CS measurement operator is required to satisfy a restricted injectivity condition. We further show that these conditions are not only sufficient but even necessary to obtain linear convergence.

On a new simple algorithm to compute the resolvents

Resolvents of operators are the core of many fundamental algorithms used in optimization. However their computation is in general difficult except for very particular operators. In the paper we provide a new simple algorithm with linear convergence rate to compute the resolvents for the class of operators which can be decomposed as a sum of a maximally monotone operator with a computable resolvent and a single-valued locally Lipschitz continuous mapping.

Avoiding bad steps in Frank-Wolfe variants

The study of Frank-Wolfe (FW) variants is often complicated by the presence of different kinds of “good” and “bad” steps. In this article, we aim to simplify the convergence analysis of specific variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a non-trivial bound at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper conditions are satisfied. This algorithmic tool allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Łojasiewicz (KL) property. While the KL setting has been widely studied for proximal gradient type methods, to our knowledge, it has never been analyzed before for the Frank-Wolfe variants considered in the paper. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that such a condition is satisfied, when considering minimization problems over a polytope, by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).

Relaxed Variable Metric Primal-Dual Fixed-Point Algorithm with Applications

In this paper, a relaxed variable metric primal-dual fixed-point algorithm is proposed for solving the convex optimization problem involving the sum of two convex functions where one is differentiable with the Lipschitz continuous gradient while the other is composed of a linear operator. Based on the preconditioned forward–backward splitting algorithm, the convergence of the proposed algorithm is proved. At the same time, we show that some existing algorithms are special cases of the proposed algorithm. Furthermore, the ergodic convergence and linear convergence rates of the proposed algorithm are established under relaxed parameters. Numerical experiments on the image deblurring problems demonstrate that the proposed algorithm outperforms some existing algorithms in terms of the number of iterations.

A generalized Frank–Wolfe method with “dual averaging” for strongly convex composite optimization

We propose a simple variant of the generalized Frank–Wolfe method for solving strongly convex composite optimization problems, by introducing an additional averaging step on the dual variables. We show that in this variant, one can choose a simple constant step-size and obtain a linear convergence rate on the duality gaps. By leveraging the convergence analysis of this variant, we then analyze the local convergence rate of the logistic fictitious play algorithm, which is well-established in game theory but lacks any form of convergence rate guarantees. We show that, with high probability, this algorithm converges locally at rate O(1/t), in terms of certain expected duality gap.

Byzantine-robust variance-reduced federated learning over distributed non-i.i.d. data

We consider the federated learning problem where data on workers are not independent and identically distributed (i.i.d.). During the learning process, an unknown number of Byzantine workers may send malicious messages to the central node, leading to remarkable learning error. Most of the Byzantine-robust methods address this issue by using robust aggregation rules to aggregate the received messages, but rely on the assumption that all the regular workers have i.i.d. data, which is not the case in many federated learning applications. In light of the significance of reducing stochastic gradient noise for mitigating the effect of Byzantine attacks, we use a resampling strategy to reduce the impact of both inner variation (that describes the sample heterogeneity on every regular worker) and outer variation (that describes the sample heterogeneity among the regular workers), along with a stochastic average gradient algorithm to gradually eliminate the inner variation. The variance-reduced messages are then aggregated with a robust geometric median operator. We prove that the proposed method reaches a neighborhood of the optimal solution at a linear convergence rate and the learning error is determined by the number of Byzantine workers. Numerical experiments corroborate the theoretical results and show that the proposed method outperforms the state-of-the-arts in the non-i.i.d. setting.

Stochastic privacy-preserving methods for nonconvex sparse learning

Sparse learning is essential in mining high-dimensional data. Iterative hard thresholding (IHT) methods are effective for optimizing nonconvex objectives for sparse learning. However, IHT methods are vulnerable to adversary attacks that infer sensitive data. Although pioneering works attempted to relieve such vulnerability, they confront the issue of high computational cost for large-scale problems. We propose two differentially private stochastic IHT: one based on the stochastic gradient descent method (DP-SGD-HT) and the other based on the stochastically controlled stochastic gradient method (DP-SCSG-HT). The DP-SGD-HT method perturbs stochastic gradients with small Gaussian noise rather than full gradients, which are computationally expensive. As a result, computational complexity is reduced from O(nlog(n)) to a lower O(blog(n)), where n is the sample size and b is the mini-batch size used to compute stochastic gradients. The DP-SCSG-HT method further perturbs the stochastic gradients controlled by large-batch snapshot gradients to reduce stochastic gradient variance. We prove that both algorithms guarantee differential privacy and have linear convergence rates with estimation bias. A utility analysis examines the relationship between convergence rate and the level of perturbation, yielding the best-known utility bound for nonconvex sparse optimization. Extensive experiments show that our algorithms outperform existing methods.

Dual Certificates and Efficient Rational Sum-of-Squares Decompositions for Polynomial Optimization over Compact Sets

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. For a theoretical application, we give a short new proof of Powers's theorems on the existence of rational WSOS certificates of positive polynomials. For a computational application, we show that exact WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. We also present an algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.

A Compressed Gradient Tracking Method for Decentralized Optimization With Linear Convergence

Communication compression techniques are of growing interests for solving the decentralized optimization problem under limited communication, where the global objective is to minimize the average of local cost functions over a multiagent network using only local computation and peer-to-peer communication. In this article, we propose a novel compressed gradient tracking algorithm (C-GT) that combines gradient tracking technique with communication compression. In particular, C-GT is compatible with a general class of compression operators that unifies both unbiased and biased compressors. We show that C-GT inherits the advantages of gradient tracking-based algorithms and achieves linear convergence rate for strongly convex and smooth objective functions. Numerical examples complement the theoretical findings and demonstrate the efficiency and flexibility of the proposed algorithm.