1. Tuomas Sandholm Carnegie Mellon University Computer Science Department
Neural Networks
Tuomas Sandholm
Carnegie Mellon University
Computer Science Department

2. How the brain works
Synaptic connections exhibit long-term changes in the connection strengths based on patterns seen

3. Comparing brains with digital computers
Parallelism
Graceful degradation
Inductive learning

4. ANN
(software/hardware,
synchronous/asynchronous)
Notation

5. Single unit (neuron) of an artificial neural network

6. Activation Functions
Where W0,i = t and a0= -1  fixed

7. Boolean gates can be simulated by units with a step function
AND OR NOT
g is a step function

8. Topologies Feed-forward vs. recurrent
Recurrent networks have state (activations from previous time steps have to be remembered): Short-term memory.

9. 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 #units training examples

10. 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.

11. Learning in ANNs is the process of tuning the weights
Form of nonlinear regression.

12. 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.

13. Perceptrons Implementable with one output unit Majority fn
Decision tree requires O(2n) nodes
Majority fn

14. 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?

15. Linear separability in 3D
Minority Function

16. Learning linearly separable functions
Training examples used over and over!
epoch
Err = 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

17. 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
Some
Full

18. Majority Function

19. WillWait

20. Multilayer feedforward networks
Structural credit assignment problem
Back propagation algorithm
(again, Erri=Ti-Oi)
Updating between hidden & output units.
Updating between input & hidden units:
Back propagation of the error

21. Back propagation (BP) as gradient descent search
A way of localizing the computation of the gradient to units.

22. Observations on BP as gradient descent
Minimize error  move in opposite direction of gradient
g needs to be differentiable
Cannot use sign fn or step fn
Use e.g. sigmoid g’=g(1-g)
Gradient taken wrt. one training example at a time

23. ANN learning curve
WillWait problem

24. WillWait Problem

25. 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]

26. Efficiency of BP Using is fast Training is slow Epoch takes
May need exponentially many epochs in #inputs

27. 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

28. Summary of representation capabilities (model class) of different supervised learning methods
3-layer feedforward ANN
Decision Tree
Perceptron
K-Nearest neighbor
Version space