AN INTRODUCTION TO NEURAL NETWORKS. Scott Kuindersma November 12, 2009

Transcription

1 AN INTRODUCTION TO NEURAL NETWORKS Scott Kuindersma November 12, 2009

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3 SUPERVISED LEARNING We are given some training data: We must learn a function If y is discrete, we call it classification If it is continuous, we call it regression

4 ARTIFICIAL NEURAL NETWORKS Artificial neural networks are one technique that can be used to solve supervised learning problems Very loosely inspired by biological neural networks real neural networks are much more complicated, e.g. using spike timing to encode information Neural networks consist of layers of interconnected units

5 PERCEPTRON UNIT The simplest computational neural unit is called a perceptron The input of a perceptron is a real vector x The output is either 1 or -1 Therefore, a perceptron can be applied to binary classification problems Whether or not it will be useful depends on the problem... more on this later...

6 PERCEPTRON UNIT[MITCHELL 1997]

7 SIGN FUNCTION

8 EXAMPLE Suppose we have a perceptron with 3 weights: On input x1 = 0.5, x2 = 0.0, the perceptron outputs: where x0 = 1

9 LEARNING RULE Now that we know how to calculate the output of a perceptron, we would like to find a way to modify the weights to produce output that matches the training data This is accomplished via the perceptron learning rule for an input pair where, again, x0 = 1 Loop through the training data until (nearly) all examples are classified correctly

10 MATLAB EXAMPLE

11 LIMITATIONS OF THE PERCEPTRON MODEL Can only distinguish between linearly separable classes of inputs Consider the following data:

12 PERCEPTRONS AND BOOLEAN FUNCTIONS Suppose we let the values (1,-1) correspond to true and false, respectively Can we describe a perceptron capable of computing the AND function? What about OR? NAND? NOR? XOR? Let s think about it geometrically

13 BOOLEAN FUNCS CONT D AND OR NAND NOR

14 EXAMPLE: AND Let pand(x1,x2) be the output of the perceptron with weights w0 = -0.3, w1 = 0.5, w2 = 0.5 on input x1, x2 x1 x2 pand(x1,x2)

15 XOR

16 XOR XOR cannot be represented by a perceptron, but it can be represented by a small network of perceptrons, e.g., x1 x2 x1 x2 OR NAND AND

17 PERCEPTRON CONVERGENCE The perceptron learning rule is not guaranteed to converge if the data is not linearly separable We can remedy this situation by considering linear unit and applying gradient descent The linear unit is equivalent to a perceptron without the sign function. That is, its output is given by: where x0 = 1

18 LEARNING RULE DERIVATION Goal: a weight update rule of the form First we define a suitable measure of error Typically we choose a quadratic function so we have a global minimum

19 ERROR SURFACE [MITCHELL 1997]

20 LEARNING RULE DERIVATION The learning algorithm should update each weight in the direction that minimizes the error according to our error function That is, the weight change should look something like

21 GRADIENT DESCENT

22 GRADIENT DESCENT Good: guaranteed to converge to the minimum error weight vector regardless of whether the training data are linearly separable (given that α is sufficiently small) Bad: still can only correctly classify linearly separable data

23 NETWORKS In general, many-layered networks of threshold units are capable of representing a rich variety of nonlinear decision surfaces However, to use our gradient descent approach on multi-layered networks, we must avoid the non-differentiable sign function Multiple layers of linear units can still only represent linear functions Introducing the sigmoid function...

24 SIGMOID FUNCTION

25 SIGMOID UNIT [MITCHELL 1997]

26 EXAMPLE Suppose we have a sigmoid unit k with 3 weights: On input x1 = 0.5, x2 = 0.0, the unit outputs:

27 NETWORK OF SIGMOID UNITS o 2 o 3 o output layer w hidden layer w 31 x 0 x 1 x 2 x 3

28 EXAMPLE x 0 x1 x 2

29 EXAMPLE output x 0 x1 x x x1 1 2

30 BACK-PROPAGATION Really just applying the same gradient descent approach to our network of sigmoid units We use the error function:

31 BACKPROP ALGORITHM

32 BACKPROP CONVERGENCE Unfortunately, there may exist many local minima in the error function Therefore we cannot guarantee convergence to an optimal solution as in the single linear unit case Time to convergence is also a concern Nevertheless, backprop does reasonably well in many cases

33 MATLAB EXAMPLE Quadratic decision boundary Single linear unit vs. Three-sigmoid unit backprop network... GO!

34 BACK TO ALVINN ALVINN was a 1989 project at CMU in which an autonomous vehicle learned to drive by watching a person drive ALVINN's architecture consists of a single hidden layer backpropagation network The input layer of the network is a 30x32 unit two dimensional "retina" which receives input from the vehicles video camera The output layer is a linear representation of the direction the vehicle should travel in order to keep the vehicle on the road

35 ALVINN

36 REPRESENTATIONAL POWER OF NEURAL NETWORKS Every boolean function can be represented by a network with two layers of units Every bounded continuous function can be approximated to arbitrarily accuracy by a two-layer network of sigmoid hidden units and linear output units Any function can be approximated to arbitrarily accuracy by a three layer network sigmoid hidden units and linear output units

37 READING SUGGESTIONS Mitchell, Machine Learning, Chapter 4 Russell and Norvig, AI a Modern Approach, Chapter 20