On the Convergence of the LMS Algorithm with Adaptive Learning Rate for Linear Feedforward Networks

1991 ◽  
Vol 3 (2) ◽  
pp. 226-245 ◽  
Author(s):  
Zhi-Quan Luo
Keyword(s):  
Adaptive Learning ◽  
Gradient Descent ◽  
Learning Algorithm ◽  
Learning Rate ◽  
Weight Matrix ◽  
Lms Algorithm ◽  
Optimal Weight ◽  
Training Cycle ◽  
Learning Rates ◽  
Finite Set

We consider the problem of training a linear feedforward neural network by using a gradient descent-like LMS learning algorithm. The objective is to find a weight matrix for the network, by repeatedly presenting to it a finite set of examples, so that the sum of the squares of the errors is minimized. Kohonen showed that with a small but fixed learning rate (or stepsize) some subsequences of the weight matrices generated by the algorithm will converge to certain matrices close to the optimal weight matrix. In this paper, we show that, by dynamically decreasing the learning rate during each training cycle, the sequence of matrices generated by the algorithm will converge to the optimal weight matrix. We also show that for any given ∊ > 0 the LMS algorithm, with decreasing learning rates, will generate an ∊-optimal weight matrix (i.e., a matrix of distance at most ∊ away from the optimal matrix) after O(1/∊) training cycles. This is in contrast to Ω(1/∊log 1/∊) training cycles needed to generate an ∊-optimal weight matrix when the learning rate is kept fixed. We also give a general condition for the learning rates under which the LMS learning algorithm is guaranteed to converge to the optimal weight matrix.

scholarly journals An Adaptive Learning Rate for RBFNN Using Time-Domain Feedback Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Syed Saad Azhar Ali ◽  
Muhammad Moinuddin ◽  
Kamran Raza ◽  
Syed Hasan Adil
Keyword(s):  
Neural Networks ◽  
Radial Basis Function ◽  
Adaptive Learning ◽  
Basis Function ◽  
Learning Algorithm ◽  
Time Series Prediction ◽  
Learning Rate ◽  
Theoretical Development ◽  
Feedback Analysis ◽  
Radial Basis

Radial basis function neural networks are used in a variety of applications such as pattern recognition, nonlinear identification, control and time series prediction. In this paper, the learning algorithm of radial basis function neural networks is analyzed in a feedback structure. The robustness of the learning algorithm is discussed in the presence of uncertainties that might be due to noisy perturbations at the input or to modeling mismatch. An intelligent adaptation rule is developed for the learning rate of RBFNN which gives faster convergence via an estimate of error energy while giving guarantee to thel2stability governed by the upper bounding via small gain theorem. Simulation results are presented to support our theoretical development.

scholarly journals Cognitive Electronic Jamming Decision-Making Method Based on Improved Q -Learning Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Huiqin Li ◽  
Yanling Li ◽  
Chuan He ◽  
Jianwei Zhan ◽  
Hui Zhang
Keyword(s):  
Decision Making ◽  
Gradient Descent ◽  
Learning Algorithm ◽  
Learning Rate ◽  
Stochastic Gradient Descent ◽  
Local Optima ◽  
Q Learning ◽  
Exploration Strategy ◽  
Decision Making Model ◽  
Metropolis Criterion

In this paper, a cognitive electronic jamming decision-making method based on improved Q -learning is proposed to improve the efficiency of radar jamming decision-making. First, the method adopts the simulated annealing (SA) algorithm’s Metropolis criterion to enhance the exploration strategy, balancing the contradictory relationship between exploration and utilization in the algorithm to avoid falling into local optima. At the same time, the idea of stochastic gradient descent with warm restarts (SGDR) is introduced to improve the learning rate of the algorithm, which reduces the oscillation and improves convergence speed at the later stage of the algorithm iteration. Then, a cognitive electronic jamming decision-making model is constructed, and the improved Q -learning algorithm’s specific steps are given. The simulation experiment takes a multifunctional radar as an example to analyze the influence of exploration strategy and learning rate on decision-making performance. The results reveal that compared with the traditional Q -learning algorithm, the improved Q -learning algorithm proposed in this paper can fully explore and efficiently utilize and converge the results to a better solution at a faster speed. The number of iterations can be reduced to more than 50%, which proves the feasibility and effectiveness of the method applied to cognitive electronic jamming decision-making.

scholarly journals Lyapunov Stability Analysis of Gradient Descent-Learning Algorithm in Network Training

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Ahmad Banakar
Keyword(s):  
Lyapunov Stability ◽  
Gradient Descent ◽  
Learning Algorithm ◽  
Stability Theorem ◽  
Stability And Convergence ◽  
Lyapunov Stability Theorem ◽  
Network Training ◽  
Learning Rates ◽  
Lyapunov Stability Analysis ◽  
The Stability

The Lyapunov stability theorem is applied to guarantee the convergence and stability of the learning algorithm for several networks. Gradient descent learning algorithm and its developed algorithms are one of the most useful learning algorithms in developing the networks. To guarantee the stability and convergence of the learning process, the upper bound of the learning rates should be investigated. Here, the Lyapunov stability theorem was developed and applied to several networks in order to guaranty the stability of the learning algorithm.

scholarly journals Feasibility Analysis and Application of Reinforcement Learning Algorithm Based on Dynamic Parameter Adjustment

Algorithms ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 239 ◽  
Author(s):  
Menglin Li ◽  
Xueqiang Gu ◽  
Chengyi Zeng ◽  
Yuan Feng
Keyword(s):  
Machine Learning ◽  
Reinforcement Learning ◽  
Learning Algorithm ◽  
Dynamic Parameter ◽  
Learning Rate ◽  
New Method ◽  
Practical Application ◽  
Learning Rates ◽  
Dynamic Parameter Adjustment ◽  
Reinforcement Learning Algorithm

Reinforcement learning, as a branch of machine learning, has been gradually applied in the control field. However, in the practical application of the algorithm, the hyperparametric approach to network settings for deep reinforcement learning still follows the empirical attempts of traditional machine learning (supervised learning and unsupervised learning). This method ignores part of the information generated by agents exploring the environment contained in the updating of the reinforcement learning value function, which will affect the performance of the convergence and cumulative return of reinforcement learning. The reinforcement learning algorithm based on dynamic parameter adjustment is a new method for setting learning rate parameters of deep reinforcement learning. Based on the traditional method of setting parameters for reinforcement learning, this method analyzes the advantages of different learning rates at different stages of reinforcement learning and dynamically adjusts the learning rates in combination with the temporal-difference (TD) error values to achieve the advantages of different learning rates in different stages to improve the rationality of the algorithm in practical application. At the same time, by combining the Robbins–Monro approximation algorithm and deep reinforcement learning algorithm, it is proved that the algorithm of dynamic regulation learning rate can theoretically meet the convergence requirements of the intelligent control algorithm. In the experiment, the effect of this method is analyzed through the continuous control scenario in the standard experimental environment of ”Car-on-The-Hill” of reinforcement learning, and it is verified that the new method can achieve better results than the traditional reinforcement learning in practical application. According to the model characteristics of the deep reinforcement learning, a more suitable setting method for the learning rate of the deep reinforcement learning network proposed. At the same time, the feasibility of the method has been proved both in theory and in the application. Therefore, the method of setting the learning rate parameter is worthy of further development and research.

scholarly journals Sistem Pendukung Keputusan Monitoring dan Peramalan Harga Beras Kabupaten Deli Serdang, Sumatera Utara

2018 ◽  
Vol 38 (2) ◽  
pp. 208
Author(s):  
Ferlando Jubelito Simanungkalit ◽  
Benika Naibaho
Keyword(s):  
Decision Support ◽  
Support System ◽  
Adaptive Learning ◽  
Gradient Descent ◽  
Activation Function ◽  
Learning Rate ◽  
Percentage Error ◽  
Mean Square ◽  
Model Base ◽  
Forecasting Method

The goal of this research was to design a Decision Support System (DSS) to monitor and forecast the price of rice. This system was designed to help the policy makers in decision making process to stabilize the rice price. The most fitted model base of the DSS forecasting method was selected by analyzing the architecture of Artificial Neural Network (ANN). The best fitted ANN architecture was selected based on the smallest value of Mean Square Error (MSE) and Mean Absolute Percentage Error (MAPE) in training, testing, and validation. The research was done using the monthly price of rice IR64 in District Deli Serdang, North Sumatera from January 2000 to December 2015. Decision support system developing phases was used to create the best match of ANN architecture for the model base of the DSS along with the database, the knowledge base, as well as the user interface. DSS was programmed using the PHP programming and was designed in a web base to facilitate the interaction between the DSS, the system's users, and the flow of data exchange. From 73 trials unit of the ANN architecture analysis, it has been obtained that an ANN 12-1-1, purelin activation function inside the hidden layer, purelin activation function inside the output layer, traingda training algorithm (gradient descent with adaptive learning rate) and the value of learning rate was 0,1 were the best match for developing the DSS forecasting method. Furthermore, the MSE and MAPE of the training, testing and validation in a row were 0.00128 and 3.57%; 0.0319 and 5.47%; 0.0052 and 2.51%. The validation results showed that the forecasting results that has been produced by the DSS has a 90 % accuracy.ABSTRAKSistem pendukung keputusan monitoring dan peramalan harga beras dirancang untuk memberikan prediksi harga masa depan dan dukungan keputusan bagi para pembuat kebijakan dalam melakukan stabilisasi harga beras. Tujuan penelitian ini adalah merancang prototipe Sistem Pendukung Keputusan (SPK) dengan terlebih dahulu menganalisis arsitektur Jaringan Saraf Tiruan (JST) yang paling sesuai untuk digunakan sebagai metode peramalan/subsistem model SPK. Kajian dilakukan dengan menggunakan data harga bulanan komoditas beras IR64 di Kabupaten Deli Serdang, Sumatera Utara bulan Januari 2000–Desember 2015. Arsitektur model JST terbaik dipilih berdasarkan pada nilai Mean Square Error (MSE) dan Mean Absolute Percentage Error (MAPE) terkecil dari hasil pelatihan, pengujian dan validasi. Arsitektur model JST terbaik kemudian dirancang menjadi subsistem model SPK bersamaan dengan basis data, komponen pengetahuan dan tampilan antarmuka menggunakan fase-fase perancangan sistem pendukung keputusan. SPK dirancang untuk digunakan berbasis web (web base) agar memudahkan interaksi dengan pengguna (user) dan arus pertukaran data. SPK diprogram menggunakan bahasa pemrograman PHP. Dari 73 percobaan analisis arsitektur model JST yang telah dilakukan, diperoleh satu arsitektur JST dengan performa peramalan terbaik yang digunakan sebagai metode peramalan dengan arsitektur 12-1-1, fungsi aktivasi purelin pada lapisan tersembunyi, fungsi aktivasi purelin pada lapisan output, algoritma pelatihan traingda (gradient descent with adaptive learning rate) dan nilai laju pembelajaran 0,1. Nilai MSE dan MAPE dari hasil pelatihan, pengujian dan validasi berturut-turut adalah 0,00128 dan 3,57%; 0,0319 dan 5,47%; 0,0052 dan 2,51%. Hasil validasi menunjukkan bahwa hasil peramalan yang dihasilkan oleh SPK memiliki tingkat akurasi 90%.

scholarly journals Sampling Adaptive Learning Algorithm for Mobile Blind Source Separation

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Jingwen Huang ◽  
Jianshan Sun
Keyword(s):  
Mean Square Error ◽  
Blind Source Separation ◽  
Adaptive Learning ◽  
Learning Algorithm ◽  
Source Separation ◽  
Learning Rate ◽  
Mean Square ◽  
Adaptive Learning Rate ◽  
Source Signals

Learning rate plays an important role in separating a set of mixed signals through the training of an unmixing matrix, to recover an approximation of the source signals in blind source separation (BSS). To improve the algorithm in speed and exactness, a sampling adaptive learning algorithm is proposed to calculate the adaptive learning rate in a sampling way. The connection for the sampled optimal points is described through a smoothing equation. The simulation result shows that the performance of the proposed algorithm has similar Mean Square Error (MSE) to that of adaptive learning algorithm but is less time consuming.

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