Learning in Neural and Belief Networks - Feed Forward Neural Network 2001 년 3 월 28 일 안순길

Contents How the Brain works Neural Networks Perceptrons

Introduction Two view points in this chapter  Computational view points : representing function using network  Biological view points : mathematical model for brain Neuron: computing elements Neural Networks: collection of interconnected neurons

How the Brain Works Cell body (soma) :provides the support functions and structure of the cell Axon : a branching fiber which carries signals away from the neurons Synapse : converts a electrical signal into a chemical signal Dendrites : consist of more branching fibers which receive signal from other nerve cells Action potential: electrical pulse Synapse  excitatory: increasing potential  synaptic connection: plasticity  inhibitory: decreasing potential A collection of simple cells can lead to thoughts, action, and consciousness.

Comparing brains with digital computers They perform quite different tasks, have different properties Speed (in Switching speed)  computer is a million times faster  brain is a billion times faster Brain  Perform a complex task  More fault-tolerant: graceful degradation  To be trained using an inductive learning algorithm

Neural Networks NN: nodes(unit), links(has a numeric weight)  Each link has a weight  Learning : updating the weights Two computational components  linear component: input function  nonlinear component: activation function

Notation

Simple computing elements Total weighted input By applying the activation function g

Three activation function

Threshold  To cause the neuron to fire  can be replaced with an extra input weight.  The input greater than threshold, output 1  Otherwise 0

Applying neural network in Logic Gates

Network structures(I) Feed-forward networks  Unidirectional links, no cycles  DAG(directed acyclic graph)  No links between units in the same layer, no links backward to a previous layer, no links that skip a layer.  Uniformly processing from input units to output units  No internal state

input units/ output units/ hidden units Perceptron: no hidden units Multilayer networks: one or more hidden units Specific parameterized structure: fixed structure and activation function Nonlinear regression: g(nonlinear function)

Network Structures(II) Recurrent Network  The Brain similar to Recurrent Network  Brain has backward link like Recurrent  Recurrent networks have internal states stored in the activation level  Unstable, oscillate, exhibit chaotic behavior  Long computation time  Need advanced mathematical method

Network Structures(III) Examples  Hopfield networks  Bidirectional connections with symmetric weights  Associative memory: most closely resembles the new stimulus  Boltzmann machines  Stochastic(probabilitic) activation function

Optimal Network Struture(I) Too small network: in capable of representation Too big network: not generalized well  Overfitting when there are too many parameters. Feed forward NN with one hidden layer  can approximate any continuous function Feed forward NN with 2 hidden layer  can approximate any function

Optimal Network Structures(II) NERF(Network Efficiently Representable Functions)  Function that can be approximated with a small number of units  Using genetic algorithm: running the whole NN training protocol  Hill-climbing search(modifying an existing network structure)  Start with a big network: optimal brain damage  Removing weights from fully connected model  Start with a small network: tiling algorithm  Start with single unit and add subsequent units  Cross-validation techniques

Perceptrons Perceptron: single-layer, feed-forward network  Each output unit is indep. of the others  Each weight only affects one of the outputs where,

What perceptrons can represent Boolean function AND, OR, and NOT Majority function: W j =1, t=n/2 ->1 unit, n weights  In case of decision tree: O(2 n ) nodes can only represent linearly separable functions. cannot represent XOR

Examples of Perceptrons Entire input space is divided in two along a boundary defined by In Figure 19.9(a): n=2 In Figure 19.10(a): n=3

Learning linearly separable functions(I) Bad news: not many problem in this set Good news: given enough training examples, there exists a perceptron algorithm learning them. Neural network learning algorithm  Current-best-hypothesis(CBH) scheme  Hypothesis: a network defined by the current values of the weights  Initial network: randomly assigned weight in [-0.5, 0.5]  Repeat the update phase to achieve convergence  Each epoch: updating all the weights for all the examples

Learning linearly separable functions(II) Learning  The error  Err=T-O  :Rosenblatt in 1960  : learning rate Error positive  Need to increase O Error negative  Need to decrease O

Algorithm

Perceptrons(Minsky and Papert, 1969)  Limits of linearly separable functions Gradient descent search through weight space  Weight space han no local minima Difference btw. NN and other attribute-based methods such as decision trees.  Real numbers in some fixed range vs. discrete set Dealing with discrete set  Local encoding: a single input, discrete attribute values  None=0.0, Some=0.5, Full=1.0 (WillWait)  Distributed encoding: one input unit for each attribute

Example

Summary(I) Neural network is made by seeing human ’ s brain  Brain still superior to Computer in Switching Speed  More fault-tolerant Neural network  nodes(unit), links(has a numeric weight)  Each link has a weight  Learning : updating the weights  Two computational components  linear component: input function  nonlinear component: activation function

Summary(II) In this text, We only consider  Feed-forward networks  Unidirectional links, no cycles  DAG(directed acyclic graph)  No links between units in the same layer, no links backward to a previous layer, no links that skip a layer.  Uniformly processing from input units to output units  No internal state

Summary(III) Network size decides Representation Power  Overfitting when there are too many parameters. Feed forward NN with one hidden layer  can approximate any continuous function Feed forward NN with 2 hidden layer  can approximate any function

Summary(IV) Perceptron: single-layer, feed-forward network  Each output unit is indep. of the others  Each weight only affects one of the outputs  Only available in linear separable functions If Problem Space is flat, Neural Network is very available. In other words, if we make it easy in algorithm perspective, Neural network also do Basically, Back Propagation only guarantee Local Optimality in neural network