neural networks tuomas sandholm carnegie mellon university computer science department

Neural Networks

Tuomas SandholmCarnegie Mellon University

Computer Science Department

How the brain works

Synaptic connections exhibit long-term changes in the connection strengths based on patterns seen

Comparing brains with digital computers

ParallelismGraceful degradationInductive learning

Notation ANN(software/hardware,synchronous/asynchronous)

Single unit (neuron) of an artificial neural network

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Activation Functions

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Boolean gates can be simulated by units with a step function

t=1.5

W=1

W=1

t=0.5

W=1

W=1

t=-0.5W= -1

AND OR NOT

g is a step function

TopologiesFeed-forward vs. recurrent

Recurrent networks have state (activations from previous time steps have to be remembered): Short-term memory.

Hopfield network• Bidirectional symmetric (Wi,j = Wj,i) connections• g is the sign function• All units are both input and output units• Activations are 1

“Associative memory”

After training on a set of examples, a new stimulus will cause the network to settle into an activation pattern corresponding to the example in the training set that most closely resemble the new stimulus.

E.g. parts of photograph

Thrm. Can reliably store 0.138 #units training examples

Boltzman machine

• Symmetric weights• Each output is 0 or 1• Includes units that are neither input units nor output units• Stochastic g, i.e. some probability (as a fn of ini) that g=1

State transitions that resemble simulated annealing.

Approximates the configuration that best meets the training set.

Learning in ANNs is the process of tuning the weights

Form of nonlinear regression.

ANN topologyRepresentation capability vs. overfitting risk.

A feed-forward net with one hidden layer can approximate any continuous fn of the inputs.

With 2 hidden layers it can approximate any fn at all.

The #units needed in each layer may grow exponentially

Learning the topologyHill-climbing vs. genetic algorithms vs. …Removing vs. adding (nodes/connections).Compare candidates via cross-validation.

Perceptronsj

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Majority fn Implementable with one output unit

Decision tree requires O(2n) nodes

Representation capability of a perceptron

Every input can only affect the output in one direction independent of other inputs.E.g. unable to represent WillWait in the restaurant example.

Perceptrons can only represent linearly separable fns. For a given problem, does one know in advance whether it is linearly separable?

Linear separability in 3D

Minority Function

Learning linearly separable functionsTraining examples used over and over!

epochErr = T-O

Variant of perceptron learning rule.Thrm. Will learn the linearly separable target fn. (if is not too high)Intuition: gradient descent in a search space with no local optima

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Encoding for ANNsE.g. #patrons can be none, some or full

Local encoding:None=0.0, Some=0.5, Full=1.0

Distributed encoding:None 1 0 0Some 0 1 0Full 0 0 1

Majority Function

WillWait

Multilayer feedforward networksStructural credit assignment problem

Back propagation algorithm (again, Erri=Ti-Oi)Updating between hidden & output units.

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Updating between input & hidden units:

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Back propagation of the error

Back propagation (BP) as gradient descent search

A way of localizing the computation of the gradient to units.

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Observations on BP as gradient descent

1. Minimize error move in opposite direction of gradient

2. g needs to be differentiable Cannot use sign fn or step fn Use e.g. sigmoid g’=g(1-g)

3. Gradient taken wrt. one training example at a time

ANN learning curve

WillWait problem

WillWait Problem

Expressiveness of BP

2n/n hidden units needed to represent arbitrary Boolean fns of n inputs.

(such a network has O(2n) weights, and we need at least 2n bits to represent a Boolean fn)

Thrm. Any continuous fn f:[0,1]nRm

Can be implemented in a 3-layer network with 2n+1 hidden units. (activation fns take special form) [Kolmogorov]

Efficiency of BP

Using is fast

Training is slowEpoch takes May need exponentially many epochs in #inputs

|)|( wmO

More on BP…Generalization:

Good on fns where output varies smoothly with input

Sensitivity to noise:Very tolerant of noiseDoes not give a degree of certainty in the output

Transparency:Black box

Prior knowledge:Hard to “prime”

No convergence guarantees

Summary of representation capabilities (model class) of different supervised

learning methods

3-layer feedforward ANNDecision TreePerceptronK-Nearest neighborVersion space