Time-varying Step Size Research Articles

Robust variable step size least mean fourth with quotient form-based control scheme for DVR operation

ABSTRACT A robust variable step size least mean fourth with a quotient form control scheme (RVSSLMFQ) is implemented for addressing power quality concerns and delivering clean voltage to sensitive loads through a dynamic voltage restorer (DVR). This method estimates the fundamental components from distorted grid voltages, featuring an automated process with adaptive time-varying step sizes and utilizing a quotient structure to reduce mean squared error (MSE) effectively. The suggested control scheme overcomes the stability and performance issues of the least mean fourth (LMF) algorithm by improving convergence and adapting well to varying step sizes. Furthermore, RVSSLMFQ achieves remarkable results with reduced settling time and fewer peak overshoots compared to the least mean square (LMS) and LMF control algorithms. For tuning the load voltage and DC-link voltage proportional-integral (PI) controllers, the improved grey wolf optimization technique (I-GWO) is applied. The results are evaluated through MATLAB simulations.

Data-Reuse Extended NLMS Algorithm Based on Optimized Time-Varying Step-Size for System Identification

Data-Reuse Extended NLMS Algorithm Based on Optimized Time-Varying Step-Size for System Identification

Finite-time error bounds for distributed linear stochastic approximation

This paper considers a novel multi-agent linear stochastic approximation algorithm driven by Markovian noise and general consensus-type interaction, in which each agent evolves according to its local stochastic approximation process which depends on the information from its neighbors. The interconnection structure among the agents is described by a time-varying directed graph. While the convergence of consensus-based stochastic approximation algorithms when the interconnection among the agents is described by doubly stochastic matrices (at least in expectation) has been studied, less is known about the case when the interconnection matrix is simply stochastic. For any uniformly strongly connected graph sequences whose associated interaction matrices are stochastic, the paper derives finite-time bounds on the mean-square error, defined as the deviation of the output of the algorithm from the unique equilibrium point of the associated ordinary differential equation. For the case of interconnection matrices being stochastic, the equilibrium point can be any unspecified convex combination of the local equilibria of all the agents in the absence of communication. Both the cases with constant and time-varying step-sizes are considered. In the case when the convex combination is required to be a straight average and interaction between any pair of neighboring agents may be uni-directional, so that doubly stochastic matrices cannot be implemented in a distributed manner, the paper proposes a push-sum-type distributed stochastic approximation algorithm and provides its finite-time bound for the time-varying step-size case by leveraging the analysis for the consensus-type algorithm with stochastic matrices and developing novel properties of the push-sum algorithm. Distributed temporal difference learning is discussed as an illustrative application.

Finite-time convergence rates of distributed local stochastic approximation

We consider a distributed learning framework, where there are a group of agents communicating with a centralized coordinator. The goal of the agents is to find the root of an operator composed of the local operators at the agents. Such a framework models many practical problems in different areas, including those in federated learning and reinforcement learning. For solving this problem, we study the popular distributed stochastic approximation. Over a series of time epoch, each agent runs a number of local stochastic approximation steps based on its own data, whose results are then aggregated at the centralized coordinator.Existing theoretical guarantees for the finite-time performance of local stochastic approximation are studied under the common assumption that the local data at each agent is sampled i.i.d. Such an assumption may not hold in many applications, where the data are temporally dependent, for example, they are sampled from some dynamical systems. In this paper, we study the setting where the data are generated from Markov random processes, which are often used to model the systems in stochastic control and reinforcement learning. Our main contribution is to characterize the finite-time performance of the local stochastic approximation under this setting. We provide explicit formulas for the rates of this method for both constant and time-varying step sizes when the local operators are strongly monotone. Our results show that these rates are within a logarithmic factor of the comparable bounds under independent data. We also provide a number of numerical simulations to illustrate our theoretical results by applying local SA in solving problems in robust identification and reinforcement learning over multi-agent systems.

Gradient-tracking based differentially private distributed optimization with enhanced optimization accuracy

Privacy protection has become an increasingly pressing requirement in distributed optimization. However, equipping distributed optimization with differential privacy, the state-of-the-art privacy protection mechanism, will unavoidably compromise optimization accuracy. In this paper, we propose an algorithm to achieve rigorous ϵ-differential privacy in gradient-tracking based distributed optimization with enhanced optimization accuracy. More specifically, to suppress the influence of differential-privacy noise, we propose a new robust gradient-tracking based distributed optimization algorithm that allows both stepsize and the variance of injected noise to vary with time. Then, we establish a new analyzing approach that can characterize the convergence of the gradient-tracking based algorithm under both constant and time-varying stepsizes. To our knowledge, this is the first analyzing framework that can treat gradient-tracking based distributed optimization under both constant and time-varying stepsizes in a unified manner. More importantly, the new analyzing approach gives a much less conservative analytical bound on the stepsize compared with existing proof techniques for gradient-tracking based distributed optimization. We also theoretically characterize the influence of differential-privacy design on the accuracy of distributed optimization, which reveals that inter-agent interaction has a significant impact on the final optimization accuracy. Numerical simulation results confirm the theoretical predictions.

Linear convergence rate analysis of a class of exact first-order distributed methods for weight-balanced time-varying networks and uncoordinated step sizes

We analyze a class of exact distributed first order methods under a general setting on the underlying network and step-sizes. In more detail, we allow simultaneously for time-varying uncoordinated step sizes and time-varying directed weight-balanced networks, jointly connected over bounded intervals. The analyzed class of methods subsumes several existing algorithms like the unified Extra and unified DIGing (Jakovetić in IEEE Trans Signal Inf Process Netw 5(1):31–46, 2019), or the exact spectral gradient method (Jakovetić et al. in Comput Optim Appl 74:703–728, 2019) that have been analyzed before under more restrictive assumptions. Under the assumed setting, we establish R-linear convergence of the methods and present several implications that our results have on the literature. Most notably, we show that the unification strategy in Jakovetić (2019) and the spectral step-size selection strategy in Jakovetić et al. (2019) exhibit a high degree of robustness to uncoordinated time-varying step sizes and to time-varying networks.

Variance reduction on general adaptive stochastic mirror descent

In this work, we investigate the idea of variance reduction by studying its properties with general adaptive mirror descent algorithms in nonsmooth nonconvex finite-sum optimization problems. We propose a simple yet generalized framework for variance reduced adaptive mirror descent algorithms named SVRAMD and provide its convergence analysis in both the nonsmooth nonconvex problem and the P-L conditioned problem. We prove that variance reduction reduces the SFO complexity of adaptive mirror descent algorithms and thus accelerates their convergence. In particular, our general theory implies that variance reduction can be applied to algorithms using time-varying step sizes and self-adaptive algorithms such as AdaGrad and RMSProp. Moreover, the convergence rates of SVRAMD recover the best existing rates of non-adaptive variance reduced mirror descent algorithms without complicated algorithmic components. Extensive experiments in deep learning validate our theoretical findings.

A Distributed Optimization Accelerated Algorithm with Uncoordinated Time-Varying Step-Sizes in an Undirected Network

In recent years, significant progress has been made in the field of distributed optimization algorithms. This study focused on the distributed convex optimization problem over an undirected network. The target was to minimize the average of all local objective functions known by each agent while each agent communicates necessary information only with its neighbors. Based on the state-of-the-art algorithm, we proposed a novel distributed optimization algorithm, when the objective function of each agent satisfies smoothness and strong convexity. Faster convergence can be attained by utilizing Nesterov and Heavy-ball accelerated methods simultaneously, making the algorithm widely applicable to many large-scale distributed tasks. Meanwhile, the step-sizes and accelerated momentum coefficients are designed as uncoordinate, time-varying, and nonidentical, which can make the algorithm adapt to a wide range of application scenarios. Under some necessary assumptions and conditions, through rigorous theoretical analysis, a linear convergence rate was achieved. Finally, the numerical experiments over a real dataset demonstrate the superiority and efficacy of the novel algorithm compared to similar algorithms.

Diffusion Bayesian Subband Adaptive Filters for Distributed Estimation Over Sensor Networks

Sensor networks are an indispensable part of the Internet of Things (IoT), where sensors perform data acquisition and information processing tasks to obtain the parameters of interest so that IoT-based monitoring, diagnosis and other systems respond quickly to the changing conditions, instantaneous faults, etc. Distributed estimation algorithms are usually employed to estimate the parameters of interest in these IoT-based applications. However, when sensor networks have highly correlated input signals and nonstationary behavior in which the parameters of interest are time-varying, conventional distributed estimation algorithms suffer from severely degraded learning performance due to the large eigenvalue spread in the covariance matrix of the input signals and the random perturbation of the parameters of interest. To address these problems, this paper proposes two diffusion Bayesian subband adaptive filter (DBSAF) algorithms from a Bayesian learning perspective. As the highly-correlated input signal is whitened in a multiband structure and an estimate of the uncertainty in the parameters of interest is obtained by performing Bayesian inference, the proposed DBSAF algorithms are able to achieve better learning performance in comparison with the competing diffusion algorithms. The transient and steady-state mean square error performance of the proposed DBSAF algorithms are analyzed, and are verified by numerical simulations. A lower bound on the time-varying step-size is derived to maintain the optimal steady-state performance in nonstationary scenarios. A new method for the estimation of the noise variance is also proposed. Numerical simulations demonstrate the excellent learning performance of the proposed algorithms in comparison with benchmark algorithms.

Transient Analysis of the Set-Membership LMS Algorithm

Adaptive filtering algorithms, which adopt the set-membership strategy, are able to attain good steady-state performance with low computational burden. In general, such advantages are obtained by defining a bounded-error. This specification translates into a time-variant step size, chosen in each iteration according to a nonlinear function of the instantaneous error. Unfortunately, this type of nonlinear behaviour hampers the stochastic modelling of these algorithms. This work devises a novel transient analysis of the Set-Membership Least Mean Squares algorithm. Additionally, a new interpretation is advanced about the implicit optimization problem solved by the algorithm. This explanation is important since it can contribute to the design of new adaptive algorithms. The proposed theoretical analysis provides predictions for: (i) steady-state performance; (ii) transient performance; and (iii) evolution of the update probability. It is noteworthy that the latter influences the computational complexity of the algorithm. Furthermore, we perform a novel comprehensive transient analysis of a set-membership algorithm. In addition, both time-variant transfer functions and deficient-length configurations are addressed. The resulting theoretical estimates are confirmed by simulations.

IMPLEMENTATION OF ADAPTIVE BEAM STEERING FOR PHASED ARRAY ANTENNAS USING ENLMS ALGORITHM

The growing demand for wireless communications services has junction rectifier to the event of latest technologies that profit of abstraction property. This is often achieved through smart antenna sets and also the algorithms that form adaptive beams. Smart antenna systems, among alternative components, provide incentives to extend network potency and improve service performance. The least mean sq. (LMS) formula was the foremost common theme used for sensible antenna systems at intervals the belief of adjustive beam-forming algorithms. Instead of quick exploitation for standardization, the traditional square of the error vector is used. The length of the error vector is that the occurring varies of iterations. As a result of the size of the step is normalized with reference to the error, this rule is called the ENLMS formula. At intervals the ENLMS rule, the time varying step size is proportional to the square of the error vector instead of the pc file vector as at intervals the NLMS. The results that the ENLMS assembly rate and a steady state error are extra interesting than the LMS rule.

Efficient maximum power point tracking based on reweighted zero-attracting variable stepsize for grid interfaced photovoltaic systems

Maximum Power Point Tracking (MPPT) algorithms mainly govern the performances of solar photovoltaic (PV) array of smart grid systems. To optimize systems’ output power in both steady-state and transient situations of the tracking process, a framework using an individualized sparse-aware time-adjusting stepsize adaptation technique for the traditional MPPT method is presented. The objective of this framework is to adaptively predict the produced power of a PV system instead of utilizing the currently estimated power to achieve better performance with the improved tracking abilities. This framework is based on predicting the power using the previously estimated values of the PV output voltage. In addition, the adaptation process is based on a time-varying stepsize that adjusts in accordance with the value of PV voltage. This approach presents a superior performance compared to the conventional method by providing an outstanding tracking ability and maintaining a stable performance under rapid alterations of atmospheric circumstances.

Distributed constrained optimization for multi-agent systems over a directed graph with piecewise stepsize

In this paper, the distributed constrained optimization problem over a directed graph is considered. We assume that the digraph has a row-stochastic adjacency matrix, which corresponds to the case when agents assign weights to the received information individually. We present an algorithm that converges to the optimal solution even with a time-varying stepsize which is not attenuated to zero. The choice of stepsizes is relatively easy. The usable type of stepsizes is added. The diminishing or constant stepsizes can be used in our algorithm. Equality constraints and set constraints are also considered in this paper. Convergence analysis of the proposed algorithm relies on a conversion theorem between column-stochastic and row-stochastic matrices. Finally, the results of the numerical simulation are provided to verify the effectiveness of the proposed algorithm.

A Distributed Algorithm for Least Squares Solutions

In this technical note, a distributed algorithm is proposed for multiagent networks to achieve a least squares solution of a system of linear equations, in which each agent only knows part of the overall equations and communicates only with its nearby neighbors. The proposed algorithm is discrete time but does not involve small or time-varying step sizes. Given that the network is fixed, connected, and undirected, the proposed algorithm enables all agents in the network to achieve exponentially fast the same least squares solution; this is validated by simulations.

Event-triggered asynchronous distributed optimization algorithm with heterogeneous time-varying step-sizes

This paper concerns distributed convex optimization problems over time-varying undirected graphs, in which the global objective function is expressed as the sum of individual objective functions of the agents. Each agent only knows its local objective functions. To figure out such problems, an event-triggered asynchronous distributed optimization algorithm (termed as EV-ADOA) with time-varying heterogeneous step-sizes is proposed, which is suitable for undirected graphs changing over time. Under two standard assumptions on strongly convex and smoothness of local objective functions, the EV-ADOA can achieve linear convergence with a proper upper bound of the heterogeneous time-varying step-sizes. EV-ADOA with event-triggered scheme can decrease network communication, and the Zeno-like behavior strictly is excluded. The efficiency of EV-ADOA is demonstrated by experiments.

Performance enhancement of Echo Cancellation Using a Combination of Partial Update ( PU) Methods and New Variable Length LMS (NVLLMS) Algorithm

In this paper, several combination algorithms between Partial Update LMS (PU LMS) methods and previously proposed algorithm (New Variable Length LMS (NVLLMS)) have been developed. Then, the new sets of proposed algorithms were applied to an Acoustic Echo Cancellation system (AEC) in order to decrease the filter coefficients, decrease the convergence time, and enhance its performance in terms of Mean Square Error (MSE) and Echo Return Loss Enhancement (ERLE). These proposed algorithms will use the Echo Return Loss Enhancement (ERLE) to control the operation of filter's coefficient length variation. In addition, the time-varying step size is used.The total number of coefficients required was reduced by about 18% , 10% , 6%, and 16% using Periodic, Sequential, Stochastic, and M-max PU NVLLMS algorithms respectively, compared to that used by a full update method which is very important, especially in the application of mobile communication since the power consumption must be considered. In addition, the average ERLE and average Mean Square Error (MSE) for M-max PU NVLLMS are better than other proposed algorithms.

Distributed primal-dual optimisation method with uncoordinated time-varying step-sizes

ABSTRACTThis paper is concerned with the distributed optimisation problem over a multi-agent network, where the objective function is described by a sum of all the local objectives of agents. The target of agents is to collectively reach an optimal solution while minimising the global objective function. Under the assumption that the information exchange among agents is depicted by a sequence of time-varying undirected graphs, a distributed optimisation algorithm with uncoordinated time-varying step-sizes is presented, which signifies that the step-sizes of agents are not always uniform per iteration. In light of some reasonable assumptions, this paper fully conducts an explicit analysis for the convergence rate of the optimisation method. A striking feature is that the algorithm has a geometric convergence rate even if the step-sizes are time-varying and uncoordinated. Simulation results on two numerical experiments in power systems show effectiveness and performance of the proposed algorithm.

Optical communication equalized technique suitable for high-speed transmission

To solve the phase distortion and high error rate in optical signal transmission, an equalized technique is proposed, which aims to improve the constant modulus algorithm (CMA). In order to correct phase rotating and reduce the error rate with 64 quadrature amplitude modulation (QAM), the method takes the mean square error as the judgment and utilizes the time-varying step size. Simulation results demonstrate that the proposed algorithm can improve the convergence speed of constellation points, make the eye opening larger, and the signal noise ratio (SNR) can be increased by 4 dB under the same bit error rate (BER), which is efficient for the recovery of information in high-speed transmission.

An Improved Firefly Algorithm Based on Nonlinear Time-varying Step-size

An Improved Firefly Algorithm Based on Nonlinear Time-varying Step-size

Variable step‐size distributed incremental normalised LMS algorithm

A variable step-size distributed incremental normalised least mean square (DINLMS) algorithm is proposed, in which the time-varying step-size of each node in the distributed network is obtained by minimising a posterior estimation error at that node. It overcomes the drawback that fast convergence rate with high steady-state misalignment or low steady-state misalignment with slow convergence rate, which always exists in the traditional DINLMS. The simulation results show that the proposed algorithm could achieve a better performance.