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Artificial Neural Network

Yalong Li

Some slides are from http://www.cs.cmu.edu/~tom/10701_sp11/slides/NNets-701-3_24_2011_ann.pdf

Structure• Motivation• Artificial neural networks• Learning: Backpropagation Algorithm• Overfitting• Expressive Capabilities of ANNs• Summary

Some facts about our brain• Performance tends to degrade gracefully under partial damage• Learn (reorganize itself) from experience• Recovery from damage is possible• Performs massively parallel computations extremely efficiently• For example, complex visual perception occurs within less than 100 ms, that

is, 10 processing steps!(processing speed of synapses about 100hz)

• Supports our intelligence and self-awareness

Neural Networks in the Brain• Cortex, midbrain, brainstem and cerebellum

• Visual System• 10 or 11 processing stages have been identified• feedforward

• earlier processing stages (near the sensory input) to later ones (near the motor output)• feedback

Neurons and Synapses • Basic computational unit in the nervous system is the nerve cell,

or neuron.

Synaptic Learning• One way brain learn is by altering the strengths of connections between neurons, and by adding or deleting connections between neurons • LTP(long-term potentiation)

• Long-Term Potentiation:• An enduring (>1 hour) increase in synaptic efficacy that results from high-frequency stimulation of an afferent (input) pathway

• The efficacy of a synapse can change as a result of experience, providing both memory and learning through long-term potentiation. One way this happens is through release of more neurotransmitter.

• Hebbs Postulate:• "When an axon of cell A... excites[s] cell B and repeatedly or persistently takes part in firing it, some growth process or metabolic change

takes place in one or both cells so that A's efficiency as one of the cells firing B is increased.“

• Points to note about LTP:• Synapses become more or less important over time (plasticity)• LTP is based on experience• LTP is based only on local information (Hebb's postulate)

Brain

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Structure• Motivation• Artificial neural networks• Backpropagation Algorithm• Overfitting• Expressive Capabilities of ANNs• Summary

Artificial Neural Networks to learn f: X • f might be non-linear function• X (vector of) continuous and/or discrete vars• Y (vector of) continuous and/or discrete vars

• Represent by network of logistic units• Each unit is a logistic function

unit output

• MLE: train weights of all units to minimize sum of squared errors of predicted network outputs• MAP: train to minimize sum of squared errors plus weight magnitudes

Artificial Neural Networks to learn f: X

f: x-> y

• f(*) is:Nonlinear activation function, for classificationIdentity, for regression

• depends on parameters and then to allow these parameters to be adjusted, along with the coefficients {wj}

• Sigmoid function can be logistic or tanh


aj: activationsh(*): nonlinear function: activation function, determined by the nature of the data and the assumed distribution of target variables


How to define • for standard regression, is the identity:

yk = ak

• for multiple binary classification, each output unit activation is transformed using a logisticsigmoid function, so that:

yk =(ak)(a) = 1/(1+exp(-a))

• For multiclass problems, a softmax activation of the form:


Why is thatThere is a natural choice of both output unit activation function and matching error function, according to the type of problem being solved.

• Regression:linear outputs, Error = sum-of-squares error

• (Multiple independent)binary classifications:logistic sigmoid outputs, Error = cross-entropy error function

• Multiclass classification:softmax output, Error = multiclass cross-entropy error function

Two classes ?

Derivative of the error function with respect to the activation for a particularouput take the form:

A probablilistic interpretation to the network outputs is given in book PRML, M.Bishop.

Multilayer Networks of Sigmoid Units

Connectionist ModelsConsider humans:

• Neuron switching time ~.001 second• Number of neurons ~1010fs• Connections per neuron ~104-5

• Scene recognition time ~.1 second• 100 inference steps doesn’t seem like enough→ Much parallel compution

Properties of artificial neural nets(ANN’s):• Many neuron-like threshold switching units• Many weighted interconnections among units• Highly parallel, distributed process

Backpropagation Algorithm• Looks for the minium of the error function in weight space using the method of gradient descent.• The combination of weights which minimizes the error functionis considered to be a solution of the learning problem.

Sigmoid unit

Error Gradient for a Sigmoid Unit

Gradient Descent

Incremental(Stochastic) Gradient Descent

Backpropagation Algorithm(MLE)

Backpropagation Algorithm(MLE)Derivation of the BP rule:

Goal:

Error: Notation:

Backpropagation Algorithm(MLE)For ouput unit j:

)( jj ot

−𝜎 𝑗

Backpropagation Algorithm(MLE)For hidden unit j:

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