INTRODUCTION TO NEURAL NETWORKS

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1 INTRODUCTION TO NEURAL NETWORKS Pictures are taken from By Nobel Khandaker

2 Neural Networks An Introduction 2 Overview of Neural Networks Origin, Definitions, examples Basic building blocks of Neural Networks Perceptrons, Sigmoids Gradient Descent Algorithm BACKPROPAGATION Algorithm

3 What is a Neural Network? - I 3 A general, practical method for learning realvalued, discrete-valued and vector-valued functions from examples Uses of Neural Networks: Recognizing handwritten characters (Microsoft uses ANN) Recognizing spoken words Recognizing human faces Interpreting visual scenes Learning robot control strategies

4 What is a Neural Network? - II 4 Neural Network is a set of connected INPUT/OUTPUT UNITS, where each connection has a WEIGHT associated with it. Neural Network learning is also called CONNECTIONIST learning due to the connections between units. It is a case of SUPERVISED, INDUCTIVE or CLASSIFICATION learning. Neural Network learns by adjusting the weights so as to be able to correctly classify the training data and hence, after testing phase, to classify unknown data.

5 5 Example of Neural Network

6 Use of Neural Network 6 ALVINN system uses Neural Networks to steer autonomous vehicle (70 mph) The Neural Network output uses the camera input to determine steering direction

7 Invention of Neural Networks 7 Biological learning systems are built of very complex webs of interconnected neurons e.g. human brain Your brain takes about 10-1 s to recognize your mother Neural networks are built using densely interconnected set of simple units. Each unit takes a number of real valued inputs and produces a single real valued output

8 Strengths and Weaknesses of Neural 8 Networks - I Strengths Can handle against complex data (i.e., problems with many parameters) Can handle noise in the training data Prediction accuracy is generally high Neural Networks are robust, work well even when training examples contain errors Neural Networks can handle missing data well

9 Strengths and Weaknesses of NNs - II 9 Neural Network implementations are slow in the training phase A major disadvantage of neural network lies in their knowledge representation. Acquired knowledge in the form of a network units connected by weighted links is difficult for humans to interpret. This factor has motivated research in extracting the knowledge embedded in trained neural network and in representing it in forms of symbolic rules

10 Perceptron 10 Perceptron

11 Use of Perceptron 11 Say +1 represents TRUE and -1 represents FALSE How can we set the weights of a perceptron to represent AND? w 0 =-0.8, w 1 =w 2 =0.5 Name a boolean function that cannot be represented by a single perceptron XOR

12 Perceptron Training Rule - I 12 Problem: Determine the weight vector that causes the perceptron to produce correct 1output for the training examples. Several algorithms exist: Perceptron Rule Delta Rule Both of these algorithms are guaranteed to converge For perceptron rule, training examples are assumed to be linearly separable

13 Perceptron Training Rule - II 13 1 Learning will converge if: training examples are assumed to be linearly separable η is sufficiently small

14 Gradient Descent and Delta Rule - I 14 How to train perceptrons when the training examples are not linearly separable? Use the delta rule Key idea in delta rule: Use gradient descent to search the hypothesis space to find the weights that best fit the training examples

15 Gradient Descent and Delta Rule -II 15 D - set of training examples t D target output for training examples o d output of the linear unit for training example d

16 Gradient Descent and Delta Rule - III 16 The weights w o,w 1 plane represents entire hypothesis space Vertical axis represents the error E Gradient descent search determines weight vector to minimize E

17 17 Gradient Descent Algorithm

18 Multilayer Networks 18 Single perceptrons can only express linear decision surfaces Multilayer networks learned by can express nonlinear decision surfaces We need a network that can represent highly nonlinear functions We can use Sigmoid units.

19 Example of a Multilayer Network 19 Network was trained to recognize 1of 10 vowel sounds Network input consist of F1, F2 obtained from spectral analysis of sound Network prediction is the output whose value is highest Decision regions of a multilayer feed forward network

20 Sigmoid Units 20 It computes the output o as: o w x y where 1 1 e y The range of the output function is [0,1]

21 BACKPROPAGATION Algorithm - I 21 Backpropagation (training_examples, η, n in, n out, n hidden ) x t, x denotes the pair of training values denotes the vector of network input values denotes the vector of target network output values t η = learning rate n in = number of network inputs

22 BACKPROPAGATION Algorithm - II 22 Backpropagation (training_examples, η, n in, n out, n hidden ) n out = number of network outputs n hidden = number of units in the hidden layer x ji denotes the input from i to j w ji denotes the weight from unit i to j Since this is a network of multiple units, the error function is defined as: E w d D k outputs t kd o kd

23 BACKPROPAGATION Algorithm - III 23 Create a feed-forward network with n in inputs, n hidden hidden units, and n out output units Initialize all network weights to small random numbers (e.g., and 0.05) Until the termination condition is met, Do

24 BACKPROPAGATION Algorithm - IV 24 BACKPROPAGATION algorithm uses a gradient descent search through the space of possible network weights, iteratively reducing E Gradient descent may get trapped in any one of the local minimas Only guaranteed to converge to some local minimum in E However, in practice, the BACKPROPAGATION algorithm performs well Gradient descent over complex error surfaces is poorly understood

25 BACKPROPAGATION Algorithm - V 25 No methods exist to predict with certainty when local minima will cause difficulties Heuristics used to alleviate the problem of local minimas: Train multiple networks using the same data, but initialize each network with different random weights Use stochastic gradient descent Add a momentum term to the weight-update rule

26 26 Example of BACKPROPAGATION - I

27 27 Example of BACKPROPAGATION - II

28 Example of BACKPROPAGATION - III 28 Input Units Hidden Units Output A Neural Network For Simulating AND Function

29 Example of BACKPROPAGATION - III 29 The given network was trained using initial weights randomly set between (-1.0, 1.0) Learning rate η = 0.3 (x,y) = (No. of iterations of the outer loop, Sum of Squared errors)

30 Example of BACKPROPAGATION - IV 30 Evolution of hidden layers (x,y) = (No. of iterations of the outer loop, hidden unit values)

31 Example of BACKPROPAGATION - V 31 Evolution of individual weights (x,y) = (No. of iterations of the outer loop, weights from inputs to one hidden unit)

32 32 Representational Power of Feedforward Networks Set of functions that can be represented: Boolean functions Number of hidden units required grows exponentially with the number of network inputs in the worst case Continuous functions Every bounded continuous function can be approximated with a network of two layers Arbitrary functions Any arbitrary function can be approximated to an arbitrary accuracy by a network of three layers

33 Regularization - I 33 The number of input and outputs in a network are determined by the dimension of the data and the number of classes The number of hidden units (M) is a free parameter that can be adjusted to give the best predictive performance M also represents the weights and biases in the network The sub-optimum value of M could result in underfitting and over-fitting

34 Regularization - II 34 Examples of two-layer networks trained on 10 data points drawn from the sinusoidal data set

35 Regularization - III 35 How to control the complexity of a neural network to avoid over-fitting We can choose a relatively large value of M and then control the complexity by adding a regularizer term A simple regularizer is: E ~ 2 T w E w w w This function is also known as: weight decay

36 Regularization - IV 36 Problem: The simple weight decay function is inconsistent with the scaling properties of network mapping Solution: A regularizer invariant under linear transformations w W 2 w W 1 w 2 w 2 W i set of weights in the ith layer This regularizer remains unchanged with 1/ 2 1/ 2 1 a 1, 2 c 2

37 Invariances - I 37 Predictions of a classifier should remain invariant under any transformation of input variables Example: In handwritten character recognition: Each character should be classified correctly irrespective of its position (translation invariance) Each character should be classified correctly irrespective of its size (scale invariance) Neural network can learn the invariance with sufficient number of training examples

38 Invariances - II 38 What if we do not have enough training examples? Augment training set using replicas of the training pattern Example: make multiple copies of the training set of character recognition problem where each character is shifted to a different position Add a regularization term to the error function that penalizes changes in the output model when the input is transformed

39 Invariances - III 39 Synthetic warping of a handwritten digit. Top Right digits show the warped input digit (Left) using random displacement and smoothing using Gaussians of width 0.01, 30, 60. Displacement fields are shown in bottom right row.

40 Bayesian Neural Networks 40 Laplace approximation for a Bayesian neural network with 8 hidden units and a single output unit

41 Conclusion 41 What we have learned about Neural Networks? What is a Neural Network Definition, Examples Strengths and weaknesses of Neural Networks Basic building blocks Perceptrons, Sigmoids Perceptron Training Rules Delta Rules, Gradient Descent Multilayer Networks BACKPROPAGATION Algorithm description, example Regularization Invariances Bayesian Neural Networks

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