多元回归学习算法收敛速度的估计

在许多应用中,回归函数的先验信息往往不能事先获取.因此,有必要利用有效的方法学习回归函数.本文研究学习理论中的回归问题,即研究多项式空间上具有最小二乘平方损失正则学习算法的收敛速度问题.主要目的在于分析学习理论中多维回归问题的泛化误差.利用逼近论中著名Jackson算子、覆盖数理论、集合的熵数以及有关概率不等式,得到学习算法收敛速度的上、下界估计.特别地,对于满足一定条件的多元光滑回归函数,除一个对数因子外,所获的收敛速度是最优的.本文结果对研究回归学习算法的收敛性、稳定性及复杂性等有着重要的意义.     

Estimation of learning rate of least square algorithm via Jackson operator

In this paper, regression problem in learning theory is investigated by least square schemes in polynomial space. Results concerning the estimation of rate of convergence are derived. In particular, it is shown that for one variable smooth regression function, the estimation is able to achieve good rate of convergence. As a main tool in the study, the Jackson operator in approximation theory is used to estimate the rate. Finally, the obtained estimation is illustrated by applying simulated data.     

Optimal rate of the regularized regression learning algorithm

This paper studies the regularized learning algorithm associated with the least-square loss and reproducing kernel Hilbert space. The target is the error analysis for the regression problem in learning theory. The upper and lower bounds of error are simultaneously estimated, which yield the optimal learning rate. The upper bound depends on the covering number and the approximation property of the reproducing kernel Hilbert space. The lower bound lies on the entropy number of the set that includes the regression function. Also, the rate is independent of the choice of the index q of the regular term.     

A numerical method for determining monotonicity and convergence rate in iterative learning control

In iterative learning control (ILC), a lifted system representation is often used for design and analysis to determine the convergence rate of the learning algorithm. Computation of the convergence rate in the lifted setting requires construction of large N×N matrices, where N is the number of data points in an iteration. The convergence rate computation is O(N²) and is typically limited to short iteration lengths because of computational memory constraints. As an alternative approach, the implicitly restarted Arnoldi/Lanczos method (IRLM) can be used to calculate the ILC convergence rate with calculations of O(N). In this article, we show that the convergence rate calculation using IRLM can be performed using dynamic simulations rather than matrices, thereby eliminating the need for large matrix construction. In addition to faster computation, IRLM enables the calculation of the ILC convergence rate for long iteration lengths. To illustrate generality, this method is presented for multi-input multi-output, linear time-varying discrete-time systems.     

Global convergence of Oja's PCA learning algorithm with a non-zero-approaching adaptive learning rate

A non-zero-approaching adaptive learning rate is proposed to guarantee the global convergence of Oja's principal component analysis (PCA) learning algorithm. Most of the existing adaptive learning rates for Oja's PCA learning algorithm are required to approach zero as the learning step increases. However, this is not practical in many applications due to the computational round-off limitations and tracking requirements. The proposed adaptive learning rate overcomes this shortcoming. The learning rate converges to a positive constant, thus it increases the evolution rate as the learning step increases. This is different from learning rates which approach zero which slow the convergence considerably and increasingly with time. Rigorous mathematical proofs for global convergence of Oja's algorithm with the proposed learning rate are given in detail via studying the convergence of an equivalent deterministic discrete time (DDT) system. Extensive simulations are carried out to illustrate and verify the theory derived. Simulation results show that this adaptive learning rate is more suitable for Oja's PCA algorithm to be used in an online learning situation.     

The learning convergence of CMAC in cyclic learning

In this paper we discuss the learning convergence of the cerebellar model articulation controller (CMAC) in cyclic learning.We prove the following results.First,if the training samples are noiseless,the training algorithm converges if and only if the learning rate is chosen from (0,2).Second,when the training samples have noises,the learning algorithm will converge with a probability of one if the learning rate is dynamically decreased.Third,in the case with noises,with a small but fixed learning rate ε the mean square error of the weight sequences generated by the CMAC learning algorithm will be bounded by O(ε).Some simulation experiments are carried out to test these results.     

Jungck-Khan iterative scheme and higher convergence rate

In this paper, we continue the theme of analytical and numerical treatment of Jungck-type iterative schemes. In particular, we focus on a special case of Jungck-Khan iterative scheme introduced by Khan et al. [Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput. 231 (2014) 521–535] to get an insight in the strong convergence and data dependence results obtained therein. Our investigations show that this special case under different control conditions on parametric sequences provides higher convergence rate and better data dependence estimates as compared to the Jungck-Khan iterative scheme itself.     

保留精英遗传算法收敛性和收敛速度的鞅方法分析

论文引入鞅方法取代传统的马尔科夫链理论,研究保留精英遗传算法(EGA)的收敛条件和收敛速度.通过把EGA的最大适应值函数过程描述为下鞅,基于下鞅收敛定理构造使算法满足几乎处处收敛的充分条件,分析了概率1收敛充分条件与算法操作参数的关系,并计算了EGA获得全局最优解所需的最大进化代数.使用鞅方法分析遗传算法收敛性具有独特的优势,成为分析遗传算法收敛性及其性能的新方法.     

Robustness and convergence rate of a discrete-time learning control algorithm for a class of nonlinear systems

In this paper, we apply a discrete-time learning algorithm to a class of discrete-time varying nonlinear systems with affine input action and linear output having relative degree one. We investigate the robustness of the algorithm to state disturbance, measurement noise and reinitialization errors. We show that the input and the state variables are always bounded if certain conditions are met. Moreover, we shown that the input error and state error converge uniformly to zero in absence of all disturbances. In addition, we show that, after a finite number of iterations, the convergence rate is exponential in l^∞. A numerical example is added to illustrate the results. Copyright © 1999 John Wiley & Sons, Ltd.     

A modified gradient-based neuro-fuzzy learning algorithm and its convergence

Neuro-fuzzy approach is known to provide an adaptive method to generate or tune fuzzy rules for fuzzy systems. In this paper, a modified gradient-based neuro-fuzzy learning algorithm is proposed for zero-order Takagi-Sugeno inference systems. This modified algorithm, compared with conventional gradient-based neuro-fuzzy learning algorithm, reduces the cost of calculating the gradient of the error function and improves the learning efficiency. Some weak and strong convergence results for this algorithm are proved, indicating that the gradient of the error function goes to zero and the fuzzy parameter sequence goes to a fixed value, respectively. A constant learning rate is used. Some conditions for the constant learning rate to guarantee the convergence are specified. Numerical examples are provided to support the theoretical findings.     

Counterexamples to convergence theorem of maximum-entropy clustering algorithm

In this paper, we surveyed the development of maximum-entropy clustering algorithm, pointed out that the maximum-entropy clustering algorithm is not new in essence, and constructed two examples to show that the iterative sequence given by the maximum-entropy clustering algorithm may not converge to a local minimum of its objective function, but a saddle point. Based on these results, our paper shows that the convergence theorem of maximum-entropy clustering algorithm put forward by Kenneth Rose et al. does not hold in general cases.     

一般克隆选择算法的概率性收敛研究*

为了丰富克隆选择算法的理论研究并将遗传算法与克隆选择算法的收敛属性进行比较,采用与研究遗传算法类似的方法研究一般克隆选择算法概率性收敛属性,得到了克隆选择算法以一个预先定义的概率δ找到全局最优解的进化代数上界,该上界是独立于优化问题的。另外,在概率性收敛的情况下,得出了克隆选择算法与遗传算法的进化代数上界的比较结果以及相关结论。     

基于动态学习率深度神经网络的抗干扰信道编码算法

针对电子战条件下,通信信号易受压制干扰的问题,提出了一种基于动态学习率深度自编码器（dynamic learning rate deep AutoEncoder,DLr-DAE）的信道编码算法来提高系统抗压制干扰性能。首先对输入未编码信号进行预处理,将原始输入信号转换为单热矢量,随后使用训练数据样本集,用非监督学习方法训练深度自编码器,基于随机梯度下降法（SGD）更新网络参数,利用指数衰减函数,在迭代次数和网络损失函数值变化过程中动态微调学习率,减少网络迭代循环次数,避免收敛结果陷入局部最优点,从而获得面向电子战环境的信道编码深度学习网络。仿真结果表明,相比现有深度学习编码算法,该算法在取得同等误码率时,抗噪声压制干扰性能最大可提升0.74 dB。     

A proof of the convergence theorem of maximum-entropy clustering algorithm

In this paper, we point out that the counterexample constructed by Yu et al.is incorrect by using scientific computing software Sage.This means that the example cannot negate the convergence theorem of maximum entropy clustering algorithm.Furthermore, we construct an example to negate Theorem 1 in Yu's paper, and we propose Proposition 3 to prove that the limit of the iterative sequence is a local minimum of the objective function while v varies and u remains stable.Finally, we give a theoretical proof of t...     

Pi-sigma神经网络混合学习算法及收敛性分析

将一种解决函数优化问题的混合遗传算法用于Pi-sigma神经网络的训练。这种混合算法充分利用遗传算法算法的全局搜索能力,又利用了单纯型法的局部搜索能力,因此该混合遗传算法可以使Pi-sigma神经网络更快的收敛到全局最优解,而且收敛速度比遗传算法更快。实验证明了这种算法的优越性。最后还证明了该算法可以以概率1收敛到全局最优解。     

Necessary and sufficient condition for convergence of iterative learning algorithm

This correspondence is concerned with an iterative learning algorithm for MIMO linear time-varying systems. We provide a necessary and sufficient condition for the existence of a convergent algorithm. The result extends the main result in Saab (IEEE Trans. Automat. Control 40(6) (1995) 1138).     

基于自适应学习速率的改进型BP算法研究

从感知器的结构及学习规则无法执行异或问题出发,用神经网络中的BP网络来解决异或问题,消除了感知器的局限性,但BP算法在具体实现中常会出现一些问题,如：收敛速度缓慢且与其他参数存在较强的耦合关系,局部极小等。对此,从前馈神经网络的原理出发,提出了一种自适应学习速率因子方法,用于对BP算法的改进,并将改进后的算法用于二维XOR问题及多维XOR问题的学习中。仿真实验证明,改进后的算法可显著提高网络的学习速度,且学习过程具有良好的收敛性及较强的鲁棒性。     

Multiagent learning using a variable learning rate

Learning to act in a multiagent environment is a difficult problem since the normal definition of an optimal policy no longer applies. The optimal policy at any moment depends on the policies of the other agents. This creates a situation of learning a moving target. Previous learning algorithms have one of two shortcomings depending on their approach. They either converge to a policy that may not be optimal against the specific opponents' policies, or they may not converge at all. In this article we examine this learning problem in the framework of stochastic games. We look at a number of previous learning algorithms showing how they fail at one of the above criteria. We then contribute a new reinforcement learning technique using a variable learning rate to overcome these shortcomings. Specifically, we introduce the WoLF principle, “Win or Learn Fast”, for varying the learning rate. We examine this technique theoretically, proving convergence in self-play on a restricted class of iterated matrix games. We also present empirical results on a variety of more general stochastic games, in situations of self-play and otherwise, demonstrating the wide applicability of this method.