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Machine Learning Neural Networks Introduction Artificial Neural Network is based on the biological nervous system as Brain It is composed of interconnected computing units called neurons ANN like human, learn by examples Science: Model how biological neural systems, like human brain, work? How do we see? How is information stored in/retrieved from memory? How do you learn to not to touch fire? How do your eyes adapt to the amount of light in the environment? Related fields: Neuroscience, Computational Neuroscience, Psychology, Psychophysiology, Cognitive Science, Medicine, Math, Physics. 3 Brief History Old Ages: Association (William James; 1890) McCulloch-Pitts Neuron (1943,1947) Perceptrons (Rosenblatt; 1958,1962) Adaline/LMS (Widrow and Hoff; 1960) Perceptrons book (Minsky and Papert; 1969) Dark Ages: Self-organization in visual cortex (von der Malsburg; 1973) Backpropagation (Werbos, 1974) Foundations of Adaptive Resonance Theory (Grossberg; 1976) Neural Theory of Association (Amari; 1977) 3/17/2012 4 History Modern Ages: Adaptive Resonance Theory (Grossberg; 1980) Hopfield model (Hopfield; 1982, 1984) Self-organizing maps (Kohonen; 1982) Reinforcement learning (Sutton and Barto; 1983) Simulated Annealing (Kirkpatrick et al.; 1983) Boltzmann machines (Ackley, Hinton, Terrence; 1985) Backpropagation (Rumelhart, Hinton, Williams; 1986) ART-networks (Carpenter, Grossberg; 1992) Support Vector Machines 3/17/2012 5 Hebb’s Learning Law In 1949, Donald Hebb formulated William James’ principle of association into a mathematical form. • If the activation of the neurons, y1 and y2 , are both on (+1) then the weight between the two neurons grow. (Off: 0) • Else the weight between remains the same. • However, when bipolar activation {-1,+1} scheme is used, then the weights can also decrease when the activation of two neurons does not match. 3/17/2012 6 Real Neural Learning Synapses change size and strength with experience. Hebbian learning: When two connected neurons are firing at the same time, the strength of the synapse between them increases. “Neurons that fire together, wire together.” 7 Biological Neurons Human brain = tens of thousands of neurons Each neuron is connected to thousands other neurons A neuron is made of: – The soma: body of the neuron – Dendrites: filaments that provide input to the neuron – The axon: sends an output signal – Synapses: connection with other neurons – releases certain quantities of chemicals called neurotransmitters to other neurons 8 Modeling of Brain Functions 9 The biological neuron The pulses generated by the neuron travels along the axon as an electrical wave. Once these pulses reach the synapses at the end of the axon open up chemical vesicles exciting the other neuron. 10 Neural Network Simple Neuron X1 Inputs X2 Output Xn Neural Network Application •Pattern recognition can be implemented using NN •The figure can be T or H character, the network should identify each class of T or H. Navigation of a car Done by Pomerlau. The network takes inputs from a 34X36 video image and a 7X36 range finder. Output units represent “drive straight”, “turn left” or “turn right”. After training about 40 times on 1200 road images, the car drove around CMU campus at 5 km/h (using a small workstation on the car). This was almost twice the speed of any other non-NN algorithm at the time. 3/17/2012 14 Automated driving at 70 mph on a public highway Camera image 30 outputs for steering 30x32 weights 4 hidden into one out of units four hidden 30x32 pixels unit as inputs 15 Computers vs. Neural Networks “Standard” Computers Neural Networks one CPU highly parallel processing fast processing units slow processing units reliable units unreliable units static infrastructure dynamic infrastructure 16 Why Artificial Neural Networks? There are two basic reasons why we are interested in building artificial neural networks (ANNs): Technical viewpoint: Some problems such as character recognition or the prediction of future states of a system require massively parallel and adaptive processing. Biological viewpoint: ANNs can be used to replicate and simulate components of the human (or animal) brain, thereby giving us insight into natural information processing. 17 How do NNs and ANNs work? Information is transmitted as a series of electric impulses, so-called spikes. The frequency and phase of these spikes encodes the information. In biological systems, one neuron can be connected to as many as 10,000 other neurons. Usually, a neuron receives its information from other neurons in a confined area 18 An Artificial Neuron synapses x1 neuron i x2 Wi,1 Wi,2 … … xi Wi,n xn n net input signal net i (t ) wi , j (t ) x j (t ) j 1 output x i (t ) f i (neti (t )) Neural Network Input Layer Hidden 1 Hidden 2 Output Layer Network Layers The common type of ANN consists of three layers of neurons: a layer of input neurons connected to the layer of hidden neuron which is connected to a layer of output neurons. Architecture of ANN Feed-Forward networks Allow the signals to travel one way from input to output Feed-Back Networks The signals travel as loops in the network, the output is connected to the input of the network How do NNs and ANNs Learn? NNs are able to learn by adapting their connectivity patterns so that the organism improves its behavior in terms of reaching certain (evolutionary) goals. The NN achieves learning by appropriately adapting the states of its synapses. Learning Rule The learning rule modifies the weights of the connections. The learning process is divided into Supervised and Unsupervised learning Supervised Network Which means there exists an external teacher. The target is to minimization of the error between the desired and computed output Unsupervised Network Uses no external teacher and is based upon only local information. Perceptron It is a network of one neuron and hard limit transfer function X1 W1 Inputs X2 W2 f Output Wn Xn Perceptron The perceptron is given first a randomly weights vectors Perceptron is given chosen data pairs (input and desired output) Preceptron learning rule changes the weights according to the error in output Perceptron Learning Rule W new = W old + (t-a) X Where W new is the new weight W old is the old value of weight X is the input value t is the desired value of output a is the actual value of output Example Let – X1 = [0 0] and t =0 – X2 = [0 1] and t=0 – X3 = [1 0] and t=0 – X4 = [1 1] and t=1 W = [2 2] and b = -3 AND Network This example means we construct a network for AND operation. The network draw a line to separate the classes which is called Classification Perceptron Geometric View The equation below describes a (hyper-)plane in the input space consisting of real valued m-dimensional vectors. The plane splits the input space into two regions, each of them describing one class. decision region for C1 m x2 w x + w x + w >= 0 w x 1 1 2 2 0 i i w 0 0 decision i 1 boundary C1 x1 C2 w1x1 + w2x2 + w0 = 0 Problems Four one-dimensional data belonging to two classes are X = [1 -0.5 3 -2] T = [1 -1 1 -1] W = [-2.5 1.75] Boolean Functions Take in two inputs (-1 or +1) Produce one output (-1 or +1) In other contexts, use 0 and 1 Example: AND function – Produces +1 only if both inputs are +1 Example: OR function – Produces +1 if either inputs are +1 Related to the logical connectives from F.O.L. The First Neural Neural Networks AND X1 1 X1 X2 Y Y 1 1 1 1 0 0 X2 1 0 1 0 AND Function 0 0 0 Threshold(Y) = 2 Simple Networks -1 W = 1.5 x t = 0.0 W=1 y Exercises Design a neural network to recognize the problem of X1=[2 2] , t1=0 X=[1 -2], t2=1 X3=[-2 2], t3=0 X4=[-1 1], t4=1 Start with initial weights w=[0 0] and bias =0 Perceptron: Limitations The perceptron can only model linearly separable classes, like (those described by) the following Boolean functions: AND OR COMPLEMENT It cannot model the XOR. You can experiment with these functions in the Matlab practical lessons. Types of decision regions w0 w1 x1 w2 x2 0 1 w0 Network x1 w1 with a single w0 w1 x1 w2 x2 0 node x2 w2 1 L1 1 L2 Convex 1 One-hidden layer region x1 1 network that realizes L3 L4 1 -3.5 the convex region x2 1 Gaussian Neurons Another type of neurons overcomes this problem by using a Gaussian activation function: net i ( t ) 1 f i (net i (t )) e 2 fi(neti(t)) 1 0 -1 1 neti(t) Gaussian Neurons Gaussian neurons are able to realize non-linear functions. Therefore, networks of Gaussian units are in principle unrestricted with regard to the functions that they can realize. The drawback of Gaussian neurons is that we have to make sure that their net input does not exceed 1. This adds some difficulty to the learning in Gaussian networks. 41 Sigmoidal Neurons Sigmoidal neurons accept any vectors of real numbers as input, and they output a real number between 0 and 1. Sigmoidal neurons are the most common type of artificial neuron, especially in learning networks. A network of sigmoidal units with m input neurons and n output neurons realizes a network function f: Rm (0,1)n 42 Sigmoidal Neurons 1 f i (net i (t )) 1 e ( neti (t ) ) / fi(neti(t)) 1 = 1 0 -1 1 neti(t) The parameter controls the slope of the sigmoid function, while the parameter controls the horizontal offset of the function in a way similar to the threshold neurons. 43 Sigmoidal Neurons This leads to a simplified form of the sigmoid function: 1 S (net ) 1 e ( net ) We do not need a modifiable threshold , because we will use “dummy” inputs as we did for perceptrons. The choice = 1 works well in most situations and results in a very simple derivative of S(net). 44 Sigmoidal Neurons 1 S ( x) x 1 e dS ( x) e x S ' ( x) dx (1 e x ) 2 1 e x 1 1 1 x 2 x (1 e ) 1 e (1 e x ) 2 S ( x)(1 S ( x)) This result will be very useful when we develop the backpropagation algorithm. 45 Multi-layers Network Let the network of 3 layers – Input layer – Hidden layer – Output layer Each layer has different number of neurons The famous example to need the multi-layer network is XOR unction Learning rule The perceptron learning rule can not be applied to multi-layer network We use BackPropagation Algorithm in learning process Feed-forward + Backpropagation Feed-forward: – input from the features is fed forward in the network from input layer towards the output layer Backpropagation: – Method to asses the blame of errors to weights – error rate flows backwards from the output layer to the input layer (to adjust the weight in order to minimize the output error) 52 Backprop Back-propagation training algorithm illustrated: Network activation Error computation Forward Step Error propagation Backward Step Backprop adjusts the weights of the NN in order to minimize the network total mean squared error. Correlation Learning Hebbian Learning (1949): “When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes place in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.” Weight modification rule: wi,j = cxixj Eventually, the connection strength will reflect the correlation between the neurons’ outputs. Competitive Learning • Nodes compete for inputs • Node with highest activation is the winner • Winner neuron adapts its tuning (pattern of weights) even further towards the current input • Individual nodes specialize to win competition for a set of similar inputs • Process leads to most efficient neural representation of input space • Typical for unsupervised learning 55 Backpropagation Learning Similar to the Adaline, the goal of the Backpropagation learning algorithm is to modify the network’s weights so that its output vector op = (op,1, op,2, …, op,K) is as close as possible to the desired output vector dp = (dp,1, dp,2, …, dp,K) for K output neurons and input patterns p = 1, …, P. The set of input-output pairs (exemplars) {(xp, dp) | p = 1, …, P} constitutes the training set. 56 Bp Algorithm The weight change rule is ij ij .error. f ' (inputi ) new old Where is the learning factor <1 Error is the error between actual and trained value f’ is is the derivative of sigmoid function = f(1-f) Delta Rule Each observation contributes a variable amount to the output The scale of the contribution depends on the input Output errors can be blamed on the weights A least mean square (LSM) error function can be defined (ideally it should be zero) E = ½ (t – y)2 Example For the network with one neuron in input layer and one neuron in hidden layer the following values are given X=1, w1 =1, b1=-2, w2=1, b2 =1, =1 and t=1 Where X is the input value W1 is the weight connect input to hidden W2 is the weight connect hidden to output B1 and b2 are bias T is the training value Exercises Design a neural network to recognize the problem of X1=[2 2] , t1=0 X=[1 -2], t2=1 X3=[-2 2], t3=0 X4=[-1 1], t4=1 Start with initial weights w=[0 0] and bias =0 Exercises Perform one iteration of backprpgation to network of two layers. First layer has one neuron with weight 1 and bias –2. The transfer function in first layer is f=n2 The second layer has only one neuron with weight 1 and bias 1. The f in second layer is 1/n. The input to the network is x=1 and t=1 Neural Network Construct a neural network to solve the problem X1 X2 Output Initialize the 1.0 1.0 1 weights 0.75 , 0.5, 9.4 6.4 -1 and –0.6 2.5 2.1 1 8.0 7.7 -1 0.5 2.2 1 7.9 8.4 -1 7.0 7.0 -1 2.8 0.8 1 1.2 3.0 1 7.8 6.1 -1 Neural Network Construct a neural network to solve the XOR problem X1 X2 Output 1 1 0 0 0 0 1 0 1 0 1 1 Initialize the weights –7.0 , -7.0, -5.0 and –4.0 -0.5 The transfer function is linear function. -2 1 1 1 -1 -1 1 3 0.5 -0.5 Consider a transfer function as f(n) = n2. Perform one iteration of BackPropagation with a= 0.9 for neural network of two neurons in input layer and one neuron in output layer. The input values are X=[1 -1] and t = 8, the weight values between input and hidden layer are w11 = 1, w12 = - 2, w21 = 0.2, and w22 = 0.1. The weight between input and output layers are w1 = 2 and w2= -2. The bias in input layers are b1 = -1, and b2= 3. W11 X1 W1 W12 W21 X2 W2 W22 Some variations True gradient descent assumes infinitesmall learning rate (). If is too small then learning is very slow. If large, then the system's learning may never converge. Some of the possible solutions to this problem are: – Add a momentum term to allow a large learning rate. – Use a different activation function – Use a different error function – Use an adaptive learning rate – Use a good weight initialization procedure. – Use a different minimization procedure 3/17/2012 66 Problems with Local Minima Backpropagation is gradient descent search – Where the height of the hills is determined by error – But there are many dimensions to the space • One for each weight in the network Therefore backpropagation – Can find its ways into local minima One partial solution: – Random re-start: learn lots of networks • Starting with different random weight settings – Can take best network – Or can set up a “committee” of networks to categorise examples Another partial solution: Momentum Adding Momentum Imagine rolling a ball down a hill Gets stuck here Without Momentum With Momentum Momentum in Backpropagation For each weight – Remember what was added in the previous epoch In the current epoch – Add on a small amount of the previous Δ The amount is determined by – The momentum parameter, denoted α – α is taken to be between 0 and 1 How Momentum Works If direction of the weight doesn’t change – Then the movement of search gets bigger – The amount of additional extra is compounded in each epoch – May mean that narrow local minima are avoided – May also mean that the convergence rate speeds up Caution: – May not have enough momentum to get out of local minima – Also, too much momentum might carry search • Back out of the global minimum, into a local minimum Momentum Weight update becomes: wij (n+1) = (pj opi) + wij(n) The momentum parameter is chosen between 0 and 1, typically 0.9. This allows one to use higher learning rates. The momentum term filters out high frequency oscillations on the error surface. What would the learning rate be in a deep 3/17/2012 valley? 71 Problems with Overfitting Plot training example error versus test example error: Test set error is increasing! – Network is overfitting the data – Learning idiosyncrasies in data, not general principles – Big problem in Machine Learning (ANNs in particular) Avoiding Overfitting Bad idea to use training set accuracy to terminate One alternative: Use a validation set – Hold back some of the training set during training – Like a miniature test set (not used to train weights at all) – If the validation set error stops decreasing, but the training set error continues decreasing • Then it’s likely that overfitting has started to occur, so stop Another alternative: use a weight decay factor – Take a small amount off every weight after each epoch – Networks with smaller weights aren’t as highly fine tuned (overfit)

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