Step-size Parameter Research Articles

An accelerated exact distributed first-order algorithm for optimization over directed networks

Distributed optimization over networked agents has emerged as an advanced paradigm to address large-scale control, optimization, and signal-processing problems. In the last few years, the distributed first-order gradient methods have witnessed significant progress and enrichment due to the simplicity of using only the first derivatives of local functions. An exact first-order algorithm is developed in this work for distributed optimization over general directed networks with only row-stochastic weighted matrices. It employs the rescaling gradient method to address unbalanced information diffusion among agents, where the weights on the received information can be arbitrarily assigned. Moreover, uncoordinated step-sizes are employed to magnify the autonomy of agents, and an error compensation term and a heavy-ball momentum are incorporated to accelerate convergency. A linear convergence rate is rigorously proven for strongly-convex objective functions with Lipschitz continuous gradients. Explicit upper bounds of step-size and momentum parameter are provided. Finally, simulations illustrate the performance of the proposed algorithm.

A Novel NLMS Algorithm for System Identification

In this paper, we propose a novel normalized least mean squares (NLMS) algorithm for system identification applications. Our approach involves analyzing the mean squared deviation performance of the NLMS algorithm using a random walk model to select two optimal parameters, the step size and regularization parameters, for the rapid convergence of the colored input signals. We verified that the proposed algorithm exhibited faster convergence than existing algorithms, even in scenarios of sudden system changes.

A New Effective Narrowband Active Noise Control System for Accommodating Frequency Mismatch

Narrowband active noise control (NANC) has shown excellent performance in dealing with the low frequency periodic noise generated by rotating machines, such as fans, engines and power transformers. Accommodating large frequency mismatch (FM) and improving its tracking capability is required for the NANC system. The existence of FM influences the noise cancellation performance. In this paper, a frequency correction algorithm based on least mean p-power (LMP) combined with the autoregressive (AR) model is designed for the NANC system, which is simple and feasible, and has a good performance under a large step size. In the NANC system, the reference signal is handled by a delay unit and AR model, and the coefficients of the AR model are adjusted by the LMP algorithm, which fine-tunes the coefficients and offers the reference signals to the NANC system. The stability bounds for the step size parameter have also been derived in the mean sense. The designed mechanism converges fast and enhances the noise decrement. Extensive simulations are performed to demonstrate the superior performance of the proposed NANC in dealing with periodic noises.

Proportionate Robust Diffusion Recursive Least Exponential Hyperbolic Cosine Algorithm: Optimum Parameter Selection and Convergence Analysis

In this brief, an improved version of the proportionate robust diffusion recursive least exponential hyperbolic cosine algorithm is suggested. This improved version is obtained by optimally selecting the step-size parameter of this algorithm using a minimization of the squared norm of the error vector. Moreover, a complete theoretical analysis of the presented algorithm is performed. This theoretical analysis consists of mean-square convergence analysis, mean-square steady-state analysis, and a discussion on the forgetting factor parameter selection. Moreover, simulation experiments show that the improved algorithm is superior than the original algorithm and other state-of-the-art algorithms in the literature in terms of speed of convergence.

Strong convergence of a modified extragradient algorithm to solve pseudomonotone equilibrium and application to classification of diabetes mellitus

This work studies pseudomonotone equilibrium problems. The modified inertial viscosity subgradient extragradient is proposed for obtaining strong convergence, and it is proved under the assumption that the bifunction satisfies the Lipchitz-type condition. Flexible use of different stepsize parameters is offered. Moreover, the proposed algorithm is applied to solve diabetes mellitus classification problems. The algorithm’s efficiency is shown by comparing with many existing methods with 80.3% high accuracy. The training-validation loss and accuracy plots are presented to consider our good model fitting.

Optimization of Positioning Error and Time for UWB and Virtual-Real Scenes Similarity Measure Based on Error Model

To overcome the problem of large UWB positioning errors caused by obstacles and other factors in the complex environment of substations, a positioning method combining UWB and virtual-real scene similarity measures is proposed, which effectively improves the positioning accuracy of substation operation and maintenance personnel. To further improve the positioning accuracy of the operation and maintenance personnel in the complex environment of the substation, and reduce the positioning time, this paper analyzes the influence of the virtual scene image acquisition step (x, y, yaw, pitch) on the positioning error. Firstly, the mathematical relationship between the step size parameter and the positioning error as well as the positioning time is quantified using the spatial geometric relationship and probability error theory, and the spatial error model of the fusion positioning algorithm is established; then, based on the established spatial error model, the acquisition step size and the positioning error and positioning time have been optimized to make the positioning time the shortest under the conditions of allowable positioning error. The experimental results show that the spatial error model established in this paper has high accuracy and the optimization results are reliable; the spatial error model is used to optimize the acquisition step parameters, which can ensure the UWB and virtual-real scenes similarity measurement fusion positioning algorithm with positioning error less than 0.2 m (with confidence greater than 0.95) and positioning time of 0.173 s, which significantly improves the positioning speed.

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.

Zeroth-Order Non-Convex Optimization for Cooperative Multi-Agent Systems with Diminishing Step Size and Smoothing Radius

We study a class of zeroth-order distributed optimization problems, where each agent can control a partial vector and observe a local cost that depends on the joint vector of all agents, and the agents can communicate with each other with time delay. We propose and study a gradient descent-based algorithm using two-point gradient estimators with diminishing smoothing parameters and diminishing step-size and we establish the convergence rate to a first-order stationary point for general nonconvex problems. A byproduct of our proposed method with diminishing step size and smoothing parameters, as opposed to the fixed-parameter scheme, is that our proposed algorithm does not require any information regarding the local cost functions. This makes the solution appealing in practice as it allows for optimizing an unknown (black-box) global function without prior knowledge of its smoothness parameters. At the same time, the performance will adaptively match the problem instance parameters.

A modified inertial viscosity extragradient type method for equilibrium problems application to classification of diabetes mellitus: Machine learning methods

<abstract><p>Diabetes is one of the four major types of noncommunicable diseases (cardiovascular disease, diabetes, cancer and chronic respiratory diseases). It is chronic condition that occurs when the body does not produce enough insulin therefore results in raised blood sugar levels. Insulin is a hormone that regulates the blood sugar when food consumption. If the proper treatment is not received organs of the body like kidneys, nervous system and eyes can deteriorate. Therefore, it is better to predict diabetes as early as possible because lead to serious damage to many of the body's systems. In this paper, we modify extragradient method with an inertial extrapolation step and viscosity-type method to solve equilibrium problems of pseudomonotone bifunction operator in real Hilbert spaces. Strong convergence result is obtained under the assumption that the bifunction satisfies the Lipchitz-type condition. Moreover, we show choosing stepsize parameter in many ways, this shows that our algorithm is flexible using. Finally, we apply our algorithm to solve the diabetes mellitus classification in machine learning and show the algorithm's efficiency by comparing with existing algorithms.</p></abstract>

The existence of Neimark-Sacker bifurcation on a discrete-time SIS-Epidemic model incorporating logistic growth and allee effect

This article investigates the dynamical properties of a discrete time SIS-Epidemic model incorporating logistic growth rate and Allee effect. The forward Euler discretization method is employed to obtain the discrete-time model. All possible fixed points are identified including their local dynamics. Some numerical simulations by varying the step size parameter are explored to show the analytical findings, the existence of Neimark-Sacker bifurcation, and the occurrence of period-10 and 20 orbits

Optimal Control of BESS for Improved Grid Integration of the Hybrid DFIG/PV System Using Dual-Layer Adaptive Control

In view of the stochastic nature of wind and solar energy, maintaining quality power in grid-connected hybrid energy system is crucial for efficient energy utilization, especially in the presence of non-linear loads. This paper deals with the control and performance assessment of grid-integrated hybrid energy system that comprises of doubly fed induction generator (DFIG) and photovoltaic (PV) system including battery energy storage (BES) as a stand-by supply to meet utility and load demand. In order to maintain power balance and address various power quality issues encountered with hybrid renewable integration along the source and the grid/load end, a dual-layer variable-step size adaptive control is implemented for the grid-interfaced converter of the proposed system. The variable step-size based dual-layer least mean square (LMS) adaptive control functions by extracting weights of various control signals (stator and load current) where the step-size parameter adjusts according to the conditions that limit the maladjustment error and increase the rate of convergence. The DC-link voltage control realized by bidirectional BES control reduces the steady-state error and fluctuations in the voltage. An artificial bee colony (ABC) optimization is adopted for optimal gain tuning of the DC-link voltage controller as a result of which the battery performance and grid-side converter performance is enhanced with the increased power exchange to the grid. The system model is developed in MATLAB/Simulink and controller performance is evaluated for diverse operating conditions including steady-state, unbalanced non-linear load conditions, wind-speed and solar irradiation variation and specified grid power modes.

Implementation Adaptive Decision Feedback Equalizer for Time-Reversal Communication in Shallow Water Environment

Underwater wireless communications are growing very fast along with human needs for applications such as defense, state security, underwater control and monitoring systems. Until now, an acoustic signal is a practical way to achieve long distance communication in the ocean. However, the underwater acoustic channel faces many challenges including limited available bandwidth, long delays, time-variability, and Doppler-spread. These challenges can reduce the reliability of the communication system and the achievement of high data-rate becomes a challenge. Adaptive decision feedback equalization is a method to compensate for the distortion of information signals on the underwater acoustic channel. On the other hand, time reversal is an effective method of overcoming intersymbol interference (ISI) problems which is the effect of multipath phenomena in underwater channels. Spatial focusing on time reversal can reduce the co-existing system disturbances and its temporal focusing makes the received power concentrated within a few taps so that the equalizer design work becomes much simpler. The temporal focusing can also increase the transmission rate. This paper shows that the combination of time reversal and adaptive DFE (TR-DFE) has superior performance than TR and DFE itself. By modifying the step-size parameters in the adaptive DFE, the TR-DFE level of convergence and performance can be improved. The geometry-based modeling which is used proves that distance and multipath variation greatly affect the quality of time reversal communication on the underwater acoustic channel.

Convergence Acceleration via Chebyshev Step: Plausible Interpretation of Deep-Unfolded Gradient Descent

Deep unfolding is a promising deep-learning technique, whose network architecture is based on expanding the recursive structure of existing iterative algorithms. Although convergence acceleration is a remarkable advantage of deep unfolding, its theoretical aspects have not been revealed yet. The first half of this study details the theoretical analysis of the convergence acceleration in deep-unfolded gradient descent (DUGD) whose trainable parameters are step sizes. We propose a plausible interpretation of the learned step-size parameters in DUGD by introducing the principle of Chebyshev steps derived from Chebyshev polynomials. The use of Chebyshev steps in gradient descent (GD) enables us to bound the spectral radius of a matrix governing the convergence speed of GD, leading to a tight upper bound on the convergence rate. The convergence rate of GD using Chebyshev steps is shown to be asymptotically optimal, although it has no momentum terms. We also show that Chebyshev steps numerically explain the learned step-size parameters in DUGD well. In the second half of the study, %we apply the theory of Chebyshev steps and Chebyshev-periodical successive over-relaxation (Chebyshev-PSOR) is proposed for accelerating linear/nonlinear fixed-point iterations. Theoretical analysis and numerical experiments indicate that Chebyshev-PSOR exhibits significantly faster convergence for various examples such as Jacobi method and proximal gradient methods.

Gradient Temporal Difference with Momentum: Stability and Convergence

Gradient temporal difference (Gradient TD) algorithms are a popular class of stochastic approximation (SA) algorithms used for policy evaluation in reinforcement learning. Here, we consider Gradient TD algorithms with an additional heavy ball momentum term and provide choice of step size and momentum parameter that ensures almost sure convergence of these algorithms asymptotically. In doing so, we decompose the heavy ball Gradient TD iterates into three separate iterates with different step sizes. We first analyze these iterates under one-timescale SA setting using results from current literature. However, the one-timescale case is restrictive and a more general analysis can be provided by looking at a three-timescale decomposition of the iterates. In the process we provide the first conditions for stability and convergence of general three-timescale SA. We then prove that the heavy ball Gradient TD algorithm is convergent using our three-timescale SA analysis. Finally, we evaluate these algorithms on standard RL problems and report improvement in performance over the vanilla algorithms.

Improving Particle Swarm Optimization Analysis Using Differential Models

This paper employs the approach of the differential model to effectively improve the analysis of particle swarm optimization. This research uses a unified model to analyze four typical particle swarm optimization (PSO) algorithms. On this basis, the proposed approach further starts from the conversion between the differential equation model and the difference equation model and proposes a differential evolution PSO model. The simulation results of high-dimensional numerical optimization problems show that the algorithm’s performance can be greatly improved by increasing the step size parameter and using different transformation methods. This analytical method improves the performance of the PSO algorithm, and it is a feasible idea. This paper uses simple analysis to find that many algorithms are improved by using the difference model. Through simple analysis, this paper finds that many AI-related algorithms have been improved by using differential models. The PSO algorithm can be regarded as the social behavior of biological groups such as birds foraging and fish swimming. Therefore, these behaviors described above are an ongoing process and are more suitable for using differential models to improve the analysis of PSO. The simulation results of the experiment show that the differential evolution PSO algorithm based on the Runge–Kutta method can effectively avoid premature results and improve the computational efficiency of the algorithm. This research analyzes the influence of the differential model on the performance of PSO under different differenced conditions. Finally, the analytical results of the differential equation model of this paper also provide a new analytical solution.

On the exactness of the ε-constraint method for biobjective nonlinear integer programming

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.

Performance analysis of the low‐complexity adaptive channel estimation algorithms over non‐stationary multipath Rayleigh fading channels under carrier frequency offsets

The use of selective partial updating method for reducing computational complexity of adaptive algorithms has been proposed recently. However, its performance analysis has not been studied as extensively and exactly as LMS or RLS algorithms for time-varying channel estimation. This paper provides performance analysis of the low-complexity family of affine projection algorithms based on selective partial update method on the estimation of multipath Rayleigh fading channels in the presence of carrier frequency offsets (CFO) and random channel variations. The analysis is based on the calculation of the error correlation matrix of the estimation, the mean-square weight error (MSWE) and the mean-square estimation error (MSE) parameters. The analysis does not use strong assumptions like small or large step-size, and explicit closed-form expressions for the MSE of estimation are obtained only from common hypotheses in wireless communication context. In this paper, the optimum step-size parameters minimizing the MSE of estimation are analytically derived without any simplifying assumptions. For the sake of comparison with other analytical approaches, the performance of the introduced algorithms is also investigated using the energy conservation relation. Likewise, for exact performance analysis, all the moment terms that appear in closed form expressions for the MSE of estimation are evaluated. Simulations are conduced to corroborate the presented studies and show that the theoretical results agree well with the simulation results over non-stationary multipath Rayleigh fading channels.

A Variable Step Size Normalized Least-Mean-Square Algorithm Based on Data Reuse

The principal issue in acoustic echo cancellation (AEC) is to estimate the impulse response between the loudspeaker and microphone of a hands-free communication device. This application can be addressed as a system identification problem, which can be solved by using an adaptive filter. The most common one for AEC is the normalized least-mean-square (NLMS) algorithm. It is known that the overall performance of this algorithm is controlled by the value of its normalized step size parameter. In order to obtain a proper compromise between the main performance criteria (e.g., convergence rate/tracking versus accuracy/robustness), this specific term of the NLMS algorithm can be further controlled and designed as a variable parameter. This represents the main motivation behind the development of variable step size algorithms. In this paper, we propose a variable step size NLMS (VSS-NLMS) algorithm that exploits the data reuse mechanism, which aims to improve the convergence rate/tracking of the algorithm by reusing the same set of data (i.e., the input and reference signals) several times. Nevertheless, we involved an equivalent version of the data reuse NLMS, which provides the convergence modes of the algorithm. Based on this approach, a sequence of normalized step sizes can be a priori scheduled, which is advantageous in terms of the computational complexity. The simulation results in the context of AEC supported the good performance features of the proposed VSS-NLMS algorithm.

Statistical Inference for Multivariate Conditional Cumulative Distribution Function Estimation By Stochastic Approximation Method

This paper handles non-parametric estimation of a conditional cumulative distribution function (CCDF). Using a recursive approach, we set forward a multivariate recursive estimator defifined by stochastic approximation algorithm. Our basic objective is to investigate the statistical inference of our estimator and compare it with that of non-recursive Nadaraya-Watson’s estimator. From this perspective, we fifirst derive the asymptotic properties of the proposed estimator which highly depend on the choice of two parameters, the stepsize (γn) as well as the bandwidth (hn). The second generation plug-in method, a method of bandwidth selection minimizing the Mean Weighted Integrated Squared Error (MW ISE) of the estimator in reference, entails the optimal choice of the bandwidth and therefore maintains an appropriate choice of the stepsize parameter. Basically, we demonstrate that, under some conditions, the Mean Squared Error (MSE) of the proposed estimator can be smaller than the one of Nadaraya Watson’s estimator. We corroborate our theoretical results through simulation studies and two real dataset applications, namely the Insurance Company Benchmark (COIL 2000) dataset as well as the French Hospital Data of COVID-19 epidemic.

An Inertial Extragradient Direction Method with Self-Adaptive Step Size for Solving Split Minimization Problems and Its Applications to Compressed Sensing

The purpose of this work is to construct iterative methods for solving a split minimization problem using a self-adaptive step size, conjugate gradient direction, and inertia technique. We introduce and prove a strong convergence theorem in the framework of Hilbert spaces. We then demonstrate numerically how the extrapolation factor (θn) in the inertia term and a step size parameter affect the performance of our proposed algorithm. Additionally, we apply our proposed algorithms to solve the signal recovery problem. Finally, we compared our algorithm’s recovery signal quality performance to that of three previously published works.