Neural Networks
Tuomas Sandholm
Carnegie 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
Parallelism
Graceful degradation
Inductive learning
ANN
(software/hardware,
synchronous/asynchronous)
Notation
Single unit (neuron) of an
artificial neural network
ini   W j ,i a j
j
Activation Functions
n
n
j 1
j 0
ai  stept (W j ,i a j )  step0 (W j ,i a j )
Where W0,i = t and a0= -1  fixed
Boolean gates can be simulated
by units with a step function
W=1
W=1
t=1.5
W=1
AND
t=0.5
W= -1
t=-0.5
W=1
OR
g is a step function
NOT
Topologies
Feed-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 topology
Representation 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 topology
Hill-climbing vs. genetic algorithms vs. …
Removing vs. adding (nodes/connections).
Compare candidates via cross-validation.
Perceptrons
O  step0 ( W j I j )
j
Implementable with one output unit
Majority fn
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 functions
Training examples used over and over!
epoch
Err = T-O
W j  W j   * I j * Err
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
Encoding for ANNs
E.g. #patrons can be none, some or full
Local encoding:
None=0.0, Some=0.5, Full=1.0
Distributed encoding:
None
1
0
Some
0
1
Full
0
0
0
0
1
Majority Function
WillWait
Multilayer feedforward networks
Structural credit assignment problem
Back propagation algorithm
(again, Erri=Ti-Oi)
Updating between hidden & output units.
Wij  W ji   * a j * Erri * g ' (ini )
Updating between input & hidden units:
Err j   W j ,i * Erri * g ' (ini )
i
Back propagation of the error
Wkj  Wkj   * I k * Err j * g ' (in j )
Back propagation (BP) as
gradient descent search
A way of localizing the computation of the gradient to units.
1
2
(
T

O
)
 i i
2 i
1
E ( w )   (Ti  g ( W ji a j )) 2
2 i
j
E

1
(Ti  g ( Wij g ( Wkj , I k ))) 2

2 i
j
k
E
  a j (Ti  Oi ) * g ' ( W ji a j )
W ji
j
  a j (Ti  Oi ) * g ' (ini )
For hidden units we get
E
  I k * g ' (in j ) W ji * Erri * g ' (ini )
Wkj
i
  I k * g ' (in j ) * Errj
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]nRm
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 slow
Epoch takes O ( m | w |)
May need exponentially many epochs in #inputs
More on BP…
Generalization:
Good on fns where output varies smoothly with input
Sensitivity to noise:
Very tolerant of noise
Does 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 ANN
Decision Tree
Perceptron
K-Nearest neighbor
Version space