神经网络学习算法研究
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摘要
随着神经网络学科又一次研究高潮的到来,神经网络已广泛应用与科学计算,模式分类,模式提取,金融行业,国防工业,航空行业,智能控制等等。神经网络不但具备逼近任何非线性函数能力,而且网络的泛化能力很强,从而达到一种函数映射关系。正因为神经网络的以上优点,才应用十分广泛。
神经网络为建立模型提供一种很好的方法。尤其为对于复杂的,不确定性,信息量很少系统,利用神经网络能建立输入与输出的对应关系,满足相应功能,使系统设计复杂度大大减小。
学习算法是神经网络中十分重要的内容,神经网络的训练过程本质上是一个优化问题,目前利用的技术是最优化理论中技术。梯度下降法和改进的梯度下降法,是主要的训练算法。这种算法存在训练时间长,权值初始化问题,不具备全局寻优能力,样本的遗忘等等。这些问题主要是学习算法所导致的。如果能找到一个合适训练算法,以上问题都能迎刃而解。根据“没有免费午餐定理”,则需要通过增加训练时间复杂度和空间复杂度来,达到要求。
对于复杂样本和复杂系统进行训练,通常需要大量的时间,随着网络复杂,所需要的训练时间急剧上升,而且训练效果远远达到要求。因为随着网络增加所需要计算的向量梯度和Hessian矩阵需要大量时间,于是本文提出将神经网络模块化思想。通过数学公式达到函数的映射,本文对傅立叶神经网络进行建模,并提出了一种更好的学习算法。
本文通过对粒子群算法进行研究,提出了将粒子群算法和梯度下降法结合起来,即利用粒子群算法全局寻优思想,NW法初始化网络的权值和速度,利用正则化改变目标函数,利用LM算法对简单网络进行训练,利用共轭梯度法对复杂的网络进行训练,以及利用神经网络集成方法,提高神经网络泛化能力,以及对每个传递函数的系数进行优化。
本文的工作和创新之处:(1)利用一种新的思想对神经网络进行训练,以及对神经网络如何实现和参数设置提出自己看法。(2)为反馈神经网络学习算法,提供了一种新的思路。
With the development of Neural Network and its application, it is widely used in Science, technology, pattern classification, feature extraction, finance, defense, industry, aerospace, intelligence control etc. Neural Network not only has the powerful approximation ability, but also has the powerful generalization ability. So the network can set up the function. Because of them, Neural Network has been used in many circumstances.
Neural Network provides a good method for setting up Model, in particular, the system has complex factors, uncertainty, less informative. So the difficulty of system design is diminished by the Neural Network which can construct the relationship between the input and output without any theories.
The algorithm is very important in Neural Network and the training process, in essence, is the optimization process. In generally, we use the back-propagation algorithm and some improved algorithms to train the neural network. However, the back-propagation has very long training time, the initialization problem, not having global search ability, forgetting the former training samples. The main reason is the learning algorithm. According to“No free lunch theorems for optimization”, we should increase time complexity and space complexity in order to get around the problems. If we can search an efficient training method, the problems can be solved.
As to complex training set or complex system, BP algorithm need much time. With the dimension increasing, the training time needs time by quickly and the training precision is very low, because the algorithm need compute the steepest vector and Hessian matrix. We propose the modular neural network which deals with the problem. According to the Fourier transport, we also research the Fourier Neural Network and give the better algorithm in terms of training precision.
After researching on the Particle Swarm Optimization, we propose the hybrid BP-PSO which employs some technology to train the neural network. The main ideas use the search idea of PSO, use NW method to initialization network, use regularization to improve the goal function, use LM algorithm to train simple network, use conjugate algorithm to train complex system, use the neural network ensembles to improve the generalization, and optimize the coefficient of every transfer function.
The work and innovations in this paper are as follows: First, we propose the novel idea in relation to the training, how to come true the ideas and the experience of selecting parameter; Second, the paper gives the new idea of training the recurrent neural network.
神经网络为建立模型提供一种很好的方法。尤其为对于复杂的,不确定性,信息量很少系统,利用神经网络能建立输入与输出的对应关系,满足相应功能,使系统设计复杂度大大减小。
学习算法是神经网络中十分重要的内容,神经网络的训练过程本质上是一个优化问题,目前利用的技术是最优化理论中技术。梯度下降法和改进的梯度下降法,是主要的训练算法。这种算法存在训练时间长,权值初始化问题,不具备全局寻优能力,样本的遗忘等等。这些问题主要是学习算法所导致的。如果能找到一个合适训练算法,以上问题都能迎刃而解。根据“没有免费午餐定理”,则需要通过增加训练时间复杂度和空间复杂度来,达到要求。
对于复杂样本和复杂系统进行训练,通常需要大量的时间,随着网络复杂,所需要的训练时间急剧上升,而且训练效果远远达到要求。因为随着网络增加所需要计算的向量梯度和Hessian矩阵需要大量时间,于是本文提出将神经网络模块化思想。通过数学公式达到函数的映射,本文对傅立叶神经网络进行建模,并提出了一种更好的学习算法。
本文通过对粒子群算法进行研究,提出了将粒子群算法和梯度下降法结合起来,即利用粒子群算法全局寻优思想,NW法初始化网络的权值和速度,利用正则化改变目标函数,利用LM算法对简单网络进行训练,利用共轭梯度法对复杂的网络进行训练,以及利用神经网络集成方法,提高神经网络泛化能力,以及对每个传递函数的系数进行优化。
本文的工作和创新之处:(1)利用一种新的思想对神经网络进行训练,以及对神经网络如何实现和参数设置提出自己看法。(2)为反馈神经网络学习算法,提供了一种新的思路。
With the development of Neural Network and its application, it is widely used in Science, technology, pattern classification, feature extraction, finance, defense, industry, aerospace, intelligence control etc. Neural Network not only has the powerful approximation ability, but also has the powerful generalization ability. So the network can set up the function. Because of them, Neural Network has been used in many circumstances.
Neural Network provides a good method for setting up Model, in particular, the system has complex factors, uncertainty, less informative. So the difficulty of system design is diminished by the Neural Network which can construct the relationship between the input and output without any theories.
The algorithm is very important in Neural Network and the training process, in essence, is the optimization process. In generally, we use the back-propagation algorithm and some improved algorithms to train the neural network. However, the back-propagation has very long training time, the initialization problem, not having global search ability, forgetting the former training samples. The main reason is the learning algorithm. According to“No free lunch theorems for optimization”, we should increase time complexity and space complexity in order to get around the problems. If we can search an efficient training method, the problems can be solved.
As to complex training set or complex system, BP algorithm need much time. With the dimension increasing, the training time needs time by quickly and the training precision is very low, because the algorithm need compute the steepest vector and Hessian matrix. We propose the modular neural network which deals with the problem. According to the Fourier transport, we also research the Fourier Neural Network and give the better algorithm in terms of training precision.
After researching on the Particle Swarm Optimization, we propose the hybrid BP-PSO which employs some technology to train the neural network. The main ideas use the search idea of PSO, use NW method to initialization network, use regularization to improve the goal function, use LM algorithm to train simple network, use conjugate algorithm to train complex system, use the neural network ensembles to improve the generalization, and optimize the coefficient of every transfer function.
The work and innovations in this paper are as follows: First, we propose the novel idea in relation to the training, how to come true the ideas and the experience of selecting parameter; Second, the paper gives the new idea of training the recurrent neural network.
引文
[1] W.S.McCulloch and W.Pitts,”A Logical Calculus of the Ideas Immannet in Nervous Activity”,Bulletin of Mathematical Biophysics,1943(5):115-33.
[2]D.O.Hebb,The Organization of Behavior,New York:Wiley,1949,introduction and chapter 4.”The First Stage of Perception: Growth of the Assembly”, pp.60-78.
[3]F.Rosenblatt,”The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain”, Psychological Review, 1958(65):386-408.
[4]M.L.Minsky and S.A. Papert, Perceptrons, Cambridge, MA:M.I.T Press,1969.
[5]T.Kohonen,”Correlation Matrix Memories”, IEEE Transactions on Computers,1972(21).353-359.
[6]S. Grossberg,“Adaptive Pattern Classification and Universal Recoding: I.Parallel Development and Coding of Neural Feature Detectors”, Biological Cybernetics, 1976(23):121-134.
[7]J.J.Hopfield,”Neural Networks and Physical Systems with Emergent Collective Computational Abilities”, Proceedings of the National Academy of Sciences,1982(79):2554-2558.
[8]D.H.Ackley,G.E.Hinton, and T.J. Sejnowski,”A learning Algorithm for Boltzmann Machines”,Cognitive Science, 1985(9):147-169.
[9] D.E.Rumelhart,G.E. Hinton, and R.j.Willianms,”Learning Internal Representations by Error Propagation”, in Parallel Distributed Processing:Explorations in the Microstructures of cognition, 1986(1):533-536.
[10] D.S. Broomhead and D.Lowe.”Multivariable Functional Interpolation and Adaptive Networks”. Complex Systems,1998(2):321-355.
[11] Chen, S., C.F.N. Cowan, and P.M. Grant. "Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks," IEEE Transactions on Neural Networks, 1991(2): 302-309.
[12] B.Widrow and m.A.Lehr.”30 Years of Neural Networks: Perceptron,Madline and BackPropagation”. Proceedings of the IEEE,1991(78):1415-1442.
[13] Hagan, M.T., H.B. Demuth, and M.H. Beale, Neural Network Design, Boston, MA: PWS Publishing, 1996.
[14] Smith M. A local model network approach to nonlinear modelling[D]. UK: Strathclyde, 1994.
[15] Anselm B, Andrzej E, David H. Occam’s Razor. Infromation Processing Letters 1987(24):377-380.
[16] Hornik K M, Stinchcombe M,White H. Multilayer feedforward networks areuniversal approximators[J]. Neural networks,1989,2(2):359-366.
[17]李洪兴.数学神经网络—神经网络学习算法。北京师范大学学报(自然科学版),1997(1):35-42.
[18]李洪兴.数学神经网络—关于泛函联接神经网络。北京师范大学学报(自然科学版),1997(3):305-311.
[19]李洪兴.数学神经网络—离散型反向式神经网络。北京师范大学学报(自然科学版),1997(3):312-320.
[20]李洪兴.数学神经网络—某些数学方法的神经网络表示。北京师范大学学报(自然科学版),1997(4):428-434.
[21] E. Baum and D.Haussler. What size net gives valid generalization? Neural Computation, 1989(1):151:160.
[22]唐贤伦,庄陵,李银国.混合粒子群优化算法优化前向神经网络结构和参数[J]。计算机应用研究. 2007, 24:91-93.
[23]高海兵,高亮,周驰.基于粒子群优化的神经网络训练算法研究[J].电子学报.2004,9:1572-1574.
[24] Haykin, Simon. Neural Network: A Comprehensive Foundation(2nd Edition).Prentice Hall.1998.
[25] Rosenblatt, F., Principles of Neurodynamics, Washington D.C., Spartan Press, 1961.
[26] D.E. Rumelhart and J. McClelland, editors, Parallel Data Processing, Vol.1, Chapter 8, The M.I.T. Press, Cambridge, MA, 1986: 318-362.
[27]S.E.Fahlman.”Fast learning variations on back-propagation: An empirical study”. In Proceedings of the 1988 Connectionist Models Summer School.1988:38-51.
[28] M.J.D.Powell,”Radial basis functions for multivariable interpolation: A review”. In algorithms for the Approximation of Functions and Data.1987:143-167.
[29] De Jesús, O., J.M. Horn, and M.T. Hagan, "Analysis of Recurrent Network Training and Suggestions for Improvements," Proceedings of the International Joint Conference on Neural Networks, Washington, DC, July 15-19, 2001:2632-2637.
[30] Medsker, L.R., and L.C. Jain, Recurrent neural networks: design and applications, Boca Raton, FL: CRC Press, 2000.
[31] Elman, J.L., "Finding structure in time," Cognitive Science, 1990(14):179-211.
[32] Hagan, M.T., and M. Menhaj, "Training feed-forward networks with the Marquardt algorithm," IEEE Transactions on Neural Networks, 1999(4): 989-993.
[33] Battiti, R., "First and second order methods for learning: Between steepest descent and Newton's method," Neural Computation, 1992(2):141-166.
[34] Beale, E.M.L., "A derivation of conjugate gradients," in F.A. Lootsma, Ed.,Numerical methods for nonlinear optimization, London: Academic Press, 1972.
[35] Charalambous, C.,"Conjugate gradient algorithm for efficient training of artificial neural networks," IEEE Proceedings, 1992(3):301-310.
[36]Fletcher, R., and C.M. Reeves, "Function minimization by conjugate gradients," Computer Journal, 1964(7):149-154.
[37]Moller, M.F., "A scaled conjugate gradient algorithm for fast supervised learning," Neural Networks, 1993(6):525-533.
[38] Kennedy J, Eberhart R C. Particle swarm optimization. Proceedings of IEEE International Conference on Neural Networks. Piscataway, NJ: IEEE Press, 1995:1942-1948.
[39] Ben Niu , Li Li,“A Novel PSO-DE-Based Hybrid Algorithm for Global Optimization”,Lecture Notes in Artificial Intelligence, 2008:156-163.
[40] Zhang J.R., Zhang J.: A hybrid particle swarm optimization-back-propagation algorithm for feedforward neural network training. Applied Mathematics and Computation.2007:1026-1037.
[41] Ben Niu, Yunlong Zhu, XiaoXian He, Hai Shen“A Multi-swarm Optimizer Based Fuzzy Modeling Approach for Dynamic Systems Processing”, Neurocomputing, 2008:1436-1448.
[42] M.R. Sierra and C.A.C. Coello, Multi-objective particle swarm optimizers: A survey of the state-of-the-art, International Journal of Computational Intelligence Research, 2006(2): 287-308.
[43] Martinez D. OFFSET: une methode de construction incremental de reseaux de neurones multicouche et son application a la conception d’un autopilote automobile [D]. France: Universite Paul Sabatier, 1992.
[44] Ronco E, Gawthrop P. Modular Neural Networks: a state of the art[R]. Technical Report CSC95026, Center of System and Control, University of Glasgow, 1995.
[45] Ronco E, Gollee H, Gawthrop P. Modular Neural Networks and Self–Decomposition [R].Technical Report CSC-96012, Center of System and Control, Glasgow, 1997.
[46] Lu B L, Ito M, Task Decomposition and Module Combination Based on Class Relations: A Modular Neural Network for pattern classification [J]. IEEE Transactions on Neural Networks, 1999, 10 (5):1244-1256.
[47] Yong L, Xin Y. Evolving Modular Neural Networks Which Generalise Well[C].Proceeding of IEEE International Conference, 1997, 605-610.
[48] Happel B.L.M., Murre J.M.J. Design and evolution of modular neural network architectures [J]. Neural Networks, 1994, 7(6):985-1004.
[49] Blanchet O, Ramirez M. A Hybrid Approach with Modular Neural Networks and Fuzzy Logic for Time Series Prediction [C] .International Conference on Artificial Intelligence, 2006.
[50]周永权,何登旭,谢宁新.前向傅立叶神经网络系统逼近理论及学习算法[J].计算机工程与应用。2003.5:72-75.
[51]邹阿金,沈建中.傅立叶神经网络建模研究[J].湘潭大学自然科学学报,2001:23(2):23-26.
[52]文卉,王创新.基于神经网络算法的频谱分析[J].湖南师范大学学报。2006.29(1):28-51.
[53]杨旭华。神经网络及其在控制中的应用研究[D].浙江大学博士论文,2004。
[55] Shi Y, Eberhart R C. A modified particle swarmoptimizer [C]. Proceedings of the IEEE Congress on Evolutionary Computation. Piscataway, NJ: IEEE Press, 1998:303～308.
[56] Shi Y, Eberhart R C. A modified Particle Swarm Optimizer [A].Anchorage, Alaska: IEEE International Conference on Evolutionary Computation, 1998(5):69-73
[57]潘昊,侯青兰.基于粒子群优化算法的BP网络学习研究[J].计算机工程与设计.2006,16:41-43.
[58]钟慧玲,李鸿洁,李冰.基于PSO的BP训练算法[J].计算机工程与设计.2007, 28:4205-4206.
[2]D.O.Hebb,The Organization of Behavior,New York:Wiley,1949,introduction and chapter 4.”The First Stage of Perception: Growth of the Assembly”, pp.60-78.
[3]F.Rosenblatt,”The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain”, Psychological Review, 1958(65):386-408.
[4]M.L.Minsky and S.A. Papert, Perceptrons, Cambridge, MA:M.I.T Press,1969.
[5]T.Kohonen,”Correlation Matrix Memories”, IEEE Transactions on Computers,1972(21).353-359.
[6]S. Grossberg,“Adaptive Pattern Classification and Universal Recoding: I.Parallel Development and Coding of Neural Feature Detectors”, Biological Cybernetics, 1976(23):121-134.
[7]J.J.Hopfield,”Neural Networks and Physical Systems with Emergent Collective Computational Abilities”, Proceedings of the National Academy of Sciences,1982(79):2554-2558.
[8]D.H.Ackley,G.E.Hinton, and T.J. Sejnowski,”A learning Algorithm for Boltzmann Machines”,Cognitive Science, 1985(9):147-169.
[9] D.E.Rumelhart,G.E. Hinton, and R.j.Willianms,”Learning Internal Representations by Error Propagation”, in Parallel Distributed Processing:Explorations in the Microstructures of cognition, 1986(1):533-536.
[10] D.S. Broomhead and D.Lowe.”Multivariable Functional Interpolation and Adaptive Networks”. Complex Systems,1998(2):321-355.
[11] Chen, S., C.F.N. Cowan, and P.M. Grant. "Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks," IEEE Transactions on Neural Networks, 1991(2): 302-309.
[12] B.Widrow and m.A.Lehr.”30 Years of Neural Networks: Perceptron,Madline and BackPropagation”. Proceedings of the IEEE,1991(78):1415-1442.
[13] Hagan, M.T., H.B. Demuth, and M.H. Beale, Neural Network Design, Boston, MA: PWS Publishing, 1996.
[14] Smith M. A local model network approach to nonlinear modelling[D]. UK: Strathclyde, 1994.
[15] Anselm B, Andrzej E, David H. Occam’s Razor. Infromation Processing Letters 1987(24):377-380.
[16] Hornik K M, Stinchcombe M,White H. Multilayer feedforward networks areuniversal approximators[J]. Neural networks,1989,2(2):359-366.
[17]李洪兴.数学神经网络—神经网络学习算法。北京师范大学学报(自然科学版),1997(1):35-42.
[18]李洪兴.数学神经网络—关于泛函联接神经网络。北京师范大学学报(自然科学版),1997(3):305-311.
[19]李洪兴.数学神经网络—离散型反向式神经网络。北京师范大学学报(自然科学版),1997(3):312-320.
[20]李洪兴.数学神经网络—某些数学方法的神经网络表示。北京师范大学学报(自然科学版),1997(4):428-434.
[21] E. Baum and D.Haussler. What size net gives valid generalization? Neural Computation, 1989(1):151:160.
[22]唐贤伦,庄陵,李银国.混合粒子群优化算法优化前向神经网络结构和参数[J]。计算机应用研究. 2007, 24:91-93.
[23]高海兵,高亮,周驰.基于粒子群优化的神经网络训练算法研究[J].电子学报.2004,9:1572-1574.
[24] Haykin, Simon. Neural Network: A Comprehensive Foundation(2nd Edition).Prentice Hall.1998.
[25] Rosenblatt, F., Principles of Neurodynamics, Washington D.C., Spartan Press, 1961.
[26] D.E. Rumelhart and J. McClelland, editors, Parallel Data Processing, Vol.1, Chapter 8, The M.I.T. Press, Cambridge, MA, 1986: 318-362.
[27]S.E.Fahlman.”Fast learning variations on back-propagation: An empirical study”. In Proceedings of the 1988 Connectionist Models Summer School.1988:38-51.
[28] M.J.D.Powell,”Radial basis functions for multivariable interpolation: A review”. In algorithms for the Approximation of Functions and Data.1987:143-167.
[29] De Jesús, O., J.M. Horn, and M.T. Hagan, "Analysis of Recurrent Network Training and Suggestions for Improvements," Proceedings of the International Joint Conference on Neural Networks, Washington, DC, July 15-19, 2001:2632-2637.
[30] Medsker, L.R., and L.C. Jain, Recurrent neural networks: design and applications, Boca Raton, FL: CRC Press, 2000.
[31] Elman, J.L., "Finding structure in time," Cognitive Science, 1990(14):179-211.
[32] Hagan, M.T., and M. Menhaj, "Training feed-forward networks with the Marquardt algorithm," IEEE Transactions on Neural Networks, 1999(4): 989-993.
[33] Battiti, R., "First and second order methods for learning: Between steepest descent and Newton's method," Neural Computation, 1992(2):141-166.
[34] Beale, E.M.L., "A derivation of conjugate gradients," in F.A. Lootsma, Ed.,Numerical methods for nonlinear optimization, London: Academic Press, 1972.
[35] Charalambous, C.,"Conjugate gradient algorithm for efficient training of artificial neural networks," IEEE Proceedings, 1992(3):301-310.
[36]Fletcher, R., and C.M. Reeves, "Function minimization by conjugate gradients," Computer Journal, 1964(7):149-154.
[37]Moller, M.F., "A scaled conjugate gradient algorithm for fast supervised learning," Neural Networks, 1993(6):525-533.
[38] Kennedy J, Eberhart R C. Particle swarm optimization. Proceedings of IEEE International Conference on Neural Networks. Piscataway, NJ: IEEE Press, 1995:1942-1948.
[39] Ben Niu , Li Li,“A Novel PSO-DE-Based Hybrid Algorithm for Global Optimization”,Lecture Notes in Artificial Intelligence, 2008:156-163.
[40] Zhang J.R., Zhang J.: A hybrid particle swarm optimization-back-propagation algorithm for feedforward neural network training. Applied Mathematics and Computation.2007:1026-1037.
[41] Ben Niu, Yunlong Zhu, XiaoXian He, Hai Shen“A Multi-swarm Optimizer Based Fuzzy Modeling Approach for Dynamic Systems Processing”, Neurocomputing, 2008:1436-1448.
[42] M.R. Sierra and C.A.C. Coello, Multi-objective particle swarm optimizers: A survey of the state-of-the-art, International Journal of Computational Intelligence Research, 2006(2): 287-308.
[43] Martinez D. OFFSET: une methode de construction incremental de reseaux de neurones multicouche et son application a la conception d’un autopilote automobile [D]. France: Universite Paul Sabatier, 1992.
[44] Ronco E, Gawthrop P. Modular Neural Networks: a state of the art[R]. Technical Report CSC95026, Center of System and Control, University of Glasgow, 1995.
[45] Ronco E, Gollee H, Gawthrop P. Modular Neural Networks and Self–Decomposition [R].Technical Report CSC-96012, Center of System and Control, Glasgow, 1997.
[46] Lu B L, Ito M, Task Decomposition and Module Combination Based on Class Relations: A Modular Neural Network for pattern classification [J]. IEEE Transactions on Neural Networks, 1999, 10 (5):1244-1256.
[47] Yong L, Xin Y. Evolving Modular Neural Networks Which Generalise Well[C].Proceeding of IEEE International Conference, 1997, 605-610.
[48] Happel B.L.M., Murre J.M.J. Design and evolution of modular neural network architectures [J]. Neural Networks, 1994, 7(6):985-1004.
[49] Blanchet O, Ramirez M. A Hybrid Approach with Modular Neural Networks and Fuzzy Logic for Time Series Prediction [C] .International Conference on Artificial Intelligence, 2006.
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