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1、Multi-layer Perceptrons,Junying Zhang,contents,structure universal theorem MLP for classification mechanism of MLP for classification nonlinear mapping binary coding of the areas MLP for regression learning algorithm of the MLP back propagation learning algorithm heuristics in learning process,XOR a
2、nd Linear Separability Revisited,Remember that it is not possible to find weights that enable Single Layer Perceptrons to deal with non-linearly separable problems like XOR: However, Multi-Layer Perceptrons (MLPs) are able to cope with non-linearly separable problems. Historically, the problem was t
3、hat there were no learning algorithms for training MLPs. Actually, it is now quite straightforward.,Structure of an MLP,it is composed of several layers neurons within each layer are not connected ith layer is only fully connected to the (i+1)jth layer Signal is transmitted only in a feedforward man
4、ner,Structure of an MLP,Model of each neuron in the net includes A nonlinear activation function the net is nonlinear The function is smooth derivative Generally, sigmoidal function The network contains one or more layers of hidden neurons that are not part of input or output of the net enable the n
5、et to learn complex tasks,Expressive power of an MLP,Questions How many hidden layers are needed? How many units should be in a (the) hidden layer? Answers Komogorovs mapping neural network existence theorem (universal theorem),Komogorovs mapping neural network existence theorem (universal theorem),
6、Any continuous function g(x) defined on the unit hypercube can be represented in the from For properly chosen functions and It is impractical the functions and are not the simple weighted sums passed through nonlinearities favored in neural networks It tells us very little about how to find the nonl
7、inear functions based on data the central problem in network based pattern recognition those functions can be extremely complex; they are not smooth,Komogorovs mapping neural network existence theorem (universal theorem),Any continuous function g(x) can be approximated to arbitrary precision by for
8、properly chosen function f(.) when NH approaches to infinity.,MLP for classification,MLP for regression,Learning scheme,Supervised learning,Two propagation directions - Function Signal: in forward direction - Error signal: in backward direction,Learning in MLP,Objective function where,The desired ou
9、tput of the jth output neuron,The real output of the jth output neuron,Sum squared error function,Steepest descent search method Partial derivative extension,Learning rate parameter,Synaptic weight from ith neuron in k-1th layer to the jth neuron in kth layer of the network,Situation for Situation f
10、or,Back propagation learning algorithm of MLP,Updating equation where which is,Back propagation formula,For sigmoidal function f(.) we have,Speeding the learning process,Learning rate parameter,Momentum constant parameter,Heuristics for making the back-propagation algorithm perform better,Sequential
11、 versus batch update Comparison, sequential model is Computationally faster More suitable for large and highly redundant training data set Makes the search in weight space stochastic in nature Less likely to be trapped in a local minimum More difficult to establish theoretical conditions for converg
12、ence of the algorithm,Stopping criteria,the back-propagation algorithm is considered to have converged gradient vector when the Euclidean norm of the gradient vector reaches a sufficiently small gradient threshold Squired error When the absolute rate of change in the average squared error per epoch
13、is sufficiently small Generalization When generalization performance reaches a peak,Generalization performance,Overfitting downfitting,generalization performance,Cross-validation Leave-one-out,Practical Considerations for Back-Propagation Learning,How Many Hidden Units?,Different Learning Rates for
14、Different Layers?,Overview,We started by revisiting the concept of linear separability and the need for multi-layered neural networks We then saw how the Back-Propagation Learning Algorithm for multilayered networks could be derived easily from the standard gradient descent approach. We ended by looking at some practical issues that didnt arise for the single layer networks,
