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We will concentrate on two-layer, but could easily generalize to more layers zi (t) = g( Sj vij (t)xj (t) ) at time t = g ( ui (t) ) yi (t) = g( Sj wij (t)zj (t) ) at time t = g ( ai (t) ) a/u known as activation, g the activation function biases set as extra weightsįorward pass Weights are fixed during forward and backward pass at time t 1. Two-layer networks Outputs of 1st layer zi x1 x2 y1 Inputs xi Outputs yj ym 2nd layer weightswij from j to i xn 1st layer weights vij from j to i Rumelhart, Hinton and Williams (1986) (though actually invented earlier in a PhD thesis relating to economics) BP has two phases: Forward pass phase: computes ‘functional signal’, feedforward propagation of input pattern signals through network Backward pass phase: computes ‘error signal’, propagates the error backwards through network starting at output units (where the error is the difference between actual and desired output values) We therefore want to find out how weight wij affects the error ie we want:īackpropagation learning algorithm ‘BP’ Solution to credit assignment problem in MLP Rumelhart, Hinton and Williams (1986) BP has two phases: Forward pass phase: computes ‘functional signal’, feedforward propagation of input pattern signals through networkīackpropagation learning algorithm ‘BP’ Solution to credit assignment problem in MLP.Analogous to deciding how much a weight in the early layer contributes to the output and thus the error.should be altered, by how much and in which direction.In neural networks, problem relates to deciding which weights.involved in forming overall response of a learning system.Problem of assigning ‘credit’ or ‘blame’ to individual elements.There is no direct error signal for the first layers!!!!!.But with more layers how are the weights for the first 2 layers found when the error is computed for layer 3 only?.In the perceptron/single layer nets, we used gradient descent on the error function to find the correct weights: D wji = (tj - yj) xi We see that errors/updates are local to the node ie the change in the weight from node i to output j (wji) is controlled by the input that travels along the connection and the error signal from output j x1 (tj - yj) x1 ? x2 What do each of the layers do? 3rd layer can generate arbitrarily complex boundaries 1st layer draws linear boundaries 2nd layer combines the boundariesĬan also view 2nd layer as using local knowledge while 3rd layer does global With sigmoidal activation functions can show that a 3 layer net can approximate any function to arbitrary accuracy: property of Universal Approximation Proof by thinking of superposition of sigmoids Not practically useful as need arbitrarily large number of units but more of an existence proof For a 2 layer net, same is true for a 2 layer net providing function is continuous and from one finite dimensional space to anotherīP gradient descent method + multilayer networks input or output units Eachunit is a perceptron Often include bias as an extra weight.Number of hidden units per layer can be more or less than.Number of output units need not equal number of input units.Fully connected between layers Each unit is a perceptron.No direct connections between input and output layers.No direct connections between input and output layers Each unit is a perceptron.No connections within a layer Each unit is a perceptron.Three-layer networks x1 x2 Input Output xn Hidden layers (1,-1) (1,1) (-1,-1) (-1,1) This is a linearly separable problem! Since for 4 points it is always linearly separable if we want to have three points in a class +1 Minsky & Papert (1969) offered solution to XOR problem by combining perceptron unit responses using a second layer of units +1 1 3 2 XOR problem Single layer generates a linear decision boundary XOR (exclusive OR) problem 0+0=0 1+1=2=0 mod 2 1+0=1 0+1=1 Perceptron does not work here
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We will follow the latter convention 1st question: what do the extra layers gain you? Start with looking at what a single layer can’t do X1 xn NB different books refer to the above as either 4 layer (no. X1 xn Today we will introduce the MLP and the backpropagation algorithm which is used to train it MLP used to describe any general feedforward (no recurrent connections) network However, we will concentrate on nets with units arranged in layers