Neural Networks Marco Loog

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...

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

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

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

Neural Networks and Games

Neural Networks and Games

Neural Networks and Games

Neural Networks and Games

Neural Networks and Games

Neural Networks and Games

Neural Networks and Games

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

Network Structure Input units Hidden units Output units

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

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

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

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

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

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

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]

Simple Comparison

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

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

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

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

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

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

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 :

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

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

Learning NN Structures : Topic for later Lecture?

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!”

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