1. Neural Networks
Marco Loog

2. Previously in ‘Statistical Methods’...
Agents can handle uncertainty by using the methods of probability and decision theory
But first they must learn their probabilistic theories of the world from experience...

3. Previously in ‘Statistical Methods’...
Key Concepts :
Data : evidence, i.e., instantiation of one or more random variables describing the domain
Hypotheses : probabilistic theories of how the domain works

4. Previously in ‘Statistical Methods’...
Outline
Bayesian learning
Maximum a posteriori and maximum likelihood learning
Instance-based learning
Neural networks...

5. Outline Some slides from last week... Network structure Perceptrons
Multilayer Feed-Forward Neural Networks
Learning Networks?

6. Neural Networks and Games

7. Neural Networks and Games

8. Neural Networks and Games

9. Neural Networks and Games

10. Neural Networks and Games

11. Neural Networks and Games

12. Neural Networks and Games

13. So First... Neural Networks
According to Robert Hecht-Nielsen, a neural network is simply “a computing system made up of a number of simple, highly interconnected processing elements, which process information by their dynamic state response to external inputs” Simply...
We skip the biology for now
And provide the bare basics

14. Network Structure
Input units
Hidden units
Output units

15. Network Structure Feed-forward networks Recurrent networks
Feedback from output units to input

16. Feed-Forward Network
Feed-forward network = a parameterized family of nonlinear functions
g is activation function
Ws are weights to be adapted, i.e., the learning

17. Activation Functions
Often have form of a step function [a threshold] or sigmoid
N.B. thresholding = ‘degenerated’ sigmoid

18. Perceptrons Single-layer neural network Expressiveness
Perceptron with g = step function can learn AND, OR, NOT, majority, but not XOR

19. Learning in Sigmoid Perceptrons
The idea is to adjust the weights so as to minimize some measure of error on the training set
Learning is optimization of the weights
This can be done using general optimization routines for continuous spaces

20. Learning in Sigmoid Perceptrons
The idea is to adjust the weights so as to minimize some measure of error on the training set
Error measure most often used for NN is the sum of squared errors

21. Learning in Sigmoid Perceptrons
Error measure most often used for NN is the sum of squared errors
Perform optimization search by gradient descent
Weight update rule [ is learning rate]

22. Simple Comparison

23. Some Remarks
[Thresholded] perceptron learning rule converges to a consistent function for any linearly separable data set
[Sigmoid] perceptron output can be interpreted as conditional probability
Also interpretation in terms of maximum likelihood [ML] estimation possible

24. Multilayer Feed-Forward NN
Network with hidden units
Adding hidden layers enlarges the hypothesis space
Most common : single hidden layer

25. Expressiveness 2-input perceptron
2-input single-hidden-layer neural network [by ‘adding’ perceptron outputs]

26. Expressiveness
With a single, sufficiently large, hidden layer it is possible to approximate any continuous function
With two layers, discontinuous functions can be approximated as well
For particular networks it is hard to say what exactly can be represented

27. Learning in Multilayer NN
Back-propagation is used to perform weight updates in the network
Similar to perceptron learning
Major difference is that output error is clear, but how to measure the error at the nodes in the hidden layers?
Additionally, should deal with multiple outputs

28. Learning in Multilayer NN
At output layer weight-update rule is the similar as for perceptron [but then for multiple outputs i] where
Idea of back-propagation : every hidden unit contributes some fraction to the error of the output node to which it connects

29. Learning in Multilayer NN
[...] contributes some fraction to the error of the output node to which it connects
Thus errors are divided according to connection strength [or weights]
Update rule :

30. E.g.
Training curve for 100 restaurant examples : exact fit

31. Learning NN Structures?
How to find the best network structure?
Too big results in ‘lookup table’ behavior / overtraining
Too small in ‘undertraining’ / not exploiting the full expressiveness
Possibility : try different structures and validate using, for example, cross-validation
But which different structures to consider?
Start with fully connected network and remove nodes : optimal brain damage
Growing larger networks [from smaller ones], e.g. tiling and NEAT

32. Learning NN Structures : Topic for later Lecture?

33. Finally, Some Remarks
NN = possibly complex nonlinear function with many parameters that have to be tuned
Problems : slow convergence, local minima
Back-propagation explained, but other optimization schemes are possible
Perceptron can handle linear separable functions
Multilayer NN can represent any kind of function
Hard to come up with optimal network
Learning rate, initial weights, etc. have to be set
NN : not much magic there... “Keine Hekserei, nur Behändigkeit!”

34. And with that Disappointing Message...
We take a break...