Pattern Recognition and Machine Learning

5.3. Error Backpropagation 241 This update is repeated by cycling through the data either in sequence or by selecting points at ...

242 5. NEURAL NETWORKS uation of other derivatives such as the Jacobian and Hessian matrices, as we shall see later in this chap ...

5.3. Error Backpropagation 243 whereziis the activation of a unit, or input, that sends a connection to unitj, andwji is the wei ...

244 5. NEURAL NETWORKS Figure 5.7 Illustration of the calculation ofδjfor hidden unitjby backpropagation of theδ’s from those un ...

5.3. Error Backpropagation 245 For batch methods, the derivative of the total errorEcan then be obtained by repeating the above ...

246 5. NEURAL NETWORKS Next we compute theδ’s for each output unit using δk=yk−tk. (5.65) Then we backpropagate these to obtainδ ...

5.3. Error Backpropagation 247 Figure 5.8 Illustration of a modular pattern recognition system in which the Jacobian matrix can ...

248 5. NEURAL NETWORKS associated with the inputs to be propagated through the trained network in order to estimate their contri ...

5.4. The Hessian Matrix 249 activations of all of the hidden and output units in the network. Next, for each row kof the Jacobia ...

250 5. NEURAL NETWORKS An important consideration for many applications of the Hessian is the efficiency with which it can be ev ...

5.4. The Hessian Matrix 251 5.4.2 Outer product approximation When neural networks are applied to regression problems, it is com ...

252 5. NEURAL NETWORKS 5.4.3 Inverse Hessian........................ We can use the outer-product approximation to develop a com ...

5.4. The Hessian Matrix 253 Again, by using a symmetrical central differences formulation, we ensure that the residual errors ar ...

254 5. NEURAL NETWORKS Both weights in the first layer: ∂^2 En ∂w(1)ji∂wj(1)′i′ =xixi′h′′(aj′)Ijj′ ∑ k w(2)kj′δk +xixi′h′(aj′) ...

5.4. The Hessian Matrix 255 usual by summing over the contributions from each of the patterns separately. For the two-layer netw ...

256 5. NEURAL NETWORKS and acting on these with theR{·}operator, we obtain expressions for the elements of the vectorvTH R { ∂E ...

5.5. Regularization in Neural Networks 257 M=1 0 1 −1 0 1 M=3 0 1 −1 0 1 M=10 0 1 −1 0 1 Figure 5.9 Examples of two-layer networ ...

258 5. NEURAL NETWORKS take the form zj=h ( ∑ i wjixi+wj 0 ) (5.113) while the activations of the output units are given by yk= ...

5.5. Regularization in Neural Networks 259 will remain unchanged under the weight transformations provided the regularization pa ...

260 5. NEURAL NETWORKS αw 1 =1,αb 1 =1,αw 2 =1,αb 2 =1 −1 −0.5 0 0.5 1 −6 −4 −2 0 2 4 αw 1 =1,α 1 b=1,αw 2 = 10,αb 2 =1 −1 −0.5 ...