Today’s Lecture
• Administrative Details
• Learning
– Decision trees: cleanup & details
– Belief nets
– Sub-symbolic learning
• Neural networks
CS-424 Gregory Dudek
Administrativia
• Assignment 3 now available. Due in 1 week.
<http://www.cim.mcgill.ca/~dudek/424/game.pdf>
• The game for the final project has been defined. It’s description can be
found via the course home page at:
<http://www.cim.mcgill.ca/~dudek/424/game.pdf>
• Examine the game now, try it, see what good strategies might
be.
• Note: the normal late policy will not apply to the project.
– You **must** submit the electronic (executable) on time, or it may not be
evaluated (i.e. you get zero)!
– It must run on LINUX. Be certain to compile and test it on one of the linux machines
in the lab well before it’s due.
• If you are developing on another platform, regularly test on linux during
development.
CS-424 Gregory Dudek
ID3 (more…)
• Last class, discussed using entropy to select a question for
building a decision tree. This idea first developed by Quinlan
(1979) in the ID3 system, later improved resulting in C4.5
• Recap:
– Entropy for classification into sets p+ and p- is I(p+,p-)
– Consider information gain per attribute A.
• For each subtree Ai we have a few bits given by the
distribution of cases on the subtree I(pi+,pi-). To fully
classify all sub-cases, we need
Remainder(A) = i weight(i) I(pi+,pi-)
• Thus, info gain is what’s left:
Gain = I(p+,p-) - Remainder(A)
CS-424 Gregory Dudek
Final thoughts on entropy...
• Provoked by the seminar yesterday.
• Idea:
– Entropy tells you about unpredictability, or randomness.
– When selecting a question, the one with highest entropy
will carry the most information with respect to what you
knew, because the answer is hardest to predict.
– When asking if we know about something, then high
entropy is bad
– Consider the PDF of a robot’s position estimate…
More entropy means more uncertainty.
CS-424 Gregory Dudek
Training & testing
• When constructing a learning system, we what to
generalize to new example. (already discussed ad nauseam)
• How can we tell if it’s working?
– Look at how well we do with our training data.
• But…. What if we just learned a “quirk” of the data?
• Overfitting? Bad features?
– (tank classification example; table lookup)
– Look at how we do on a set of examples never used for
any training: a “test set”.
• But… what if we can’t afford the data?
– Cross validation: learner L on training set X
e(L:X) = i in X S=X-i error(L(S) on case i)2
CS-424 Gregory Dudek
Simple functions?
Is there a fixed circuit network topology that can be
used to represent
a family of functions?
Yes! Neural-like networks (a.k.a. artificial neural
networks) allow us this flexibility and more; we can
represent arbitrary families of continuous functions
using fixed topology networks.
CS-424 Gregory Dudek
Belief networks (ch. 15) - briefly
• We will cover only R&N Section 15.1 & 15.2 (briefly), and then segue to
chapter 19. You should read 15.3. This will be cursory coverage
only.
• A belief net is a formalism for describing
probabilistic relationships (a.k.a. Bayes nets).
• A graph (in fact a DAG) G(V,E). Nodes are random variables.
Directed edges indicate a node has direct influence on
another node.
• Each node has an associated conditional probability table
quantifying the effects of it’s “parents”.
• No directed cycles.
CS-424 Gregory Dudek
Why?
• Objective:
– Compute probabilities of variables of interest, query
variables
– Given observations of specific phenomena in the world,
evidence variables.
• The net you get depends on how you construct it, not
just the problem and probabilities.
• Seek compactness (fewer links, tighter clusters):
called locally structured nets.
CS-424 Gregory Dudek
• See overheads...
CS-424 Gregory Dudek
Issues
• Where do the probabilities come from?
– They can be learned or inferred from data.
• Where does the causal structure come from (the
topology)?
– It’s (sometimes) very hard to learn.
– Problem: lots of alternative topologies are possible.
What’s really cause and what’s effect?
• Did it really rain because I brought my umbrella? Can
a computer infer this (or the opposite) just from
weather data?
• Both these topics are current research areas.
CS-424 Gregory Dudek
Neural Networks?
Artificial Neural Nets
a.k.a.
Connectionist Nets
(connectionist learning)
a.k.a.
Sub-symbolic learning
a.k.a.
Perceptron learning (a special case)
CS-424 Gregory Dudek
Networks that model the brain?
• Note there is an interesting connection to Bayes
nets: it isn’t considered in the book.
– Something to reflect on.
• Idea: model intelligence withour “jumping ahead” to
symbolic representations.
• Related to earliest work on cybernetics.
CS-424 Gregory Dudek
The idealized neuron
• Artificial neural networks come in several “flavors”.
– Most of based on a simplified model of a neuron.
• A set of (many) inputs.
• One output.
• Output is a function of the sum on the inputs.
– Typical functions:
• Weighted sum
• Threshold
• Gaussian
CS-424 Gregory Dudek
Why neural nets?
• Motives:
– We wish to create systems with abilities
akin to those of the human mind.
• The mind is usually assumed to be
be a direct consequence of the
structure of the brain.
– Let’s mimic the structure of the brain!
– By using simple computing elements,
we obtain a system that might scale up
easily to parallel hardware.
– Avoids (or solves?) the key unresolved
problem of how to get from “signal
domain” to symbolic representations.
– Fault tolerance
CS-424 Gregory Dudek
CS-424 Gregory Dudek
CS-424 Gregory Dudek
Real and fake neurons
• Signals in neurons
are coded by “spike
rate”.
• In ANN’s, inputs can
be either:
–
–
–
–
0 or 1 (binary)
[0,1]
[-1,1]
R (real)
• Each input Ii has an
associated realvalued weight wi
•
Learning by changing
weights at synapses.
CS-424 Gregory Dudek
CS-424 Gregory Dudek
Brains
• The brain seems divided into functional areas
• These are often seen as analogous to modules in a
software system.
• Why would it be like this? (2 possible answers)
– Evolution: incremental improvement easier in a modular
system.
– Advantage of combining complementary solutions.
– It isn’t!
CS-424 Gregory Dudek
Inductive bias?
• Where’s the inductive bias?
–
–
–
–
In the topology and architecture of the network.
In the learning rules.
In the input and output representation.
In the initial weights.
CS-424 Gregory Dudek
Simple neural models
• Oldest ANN model is McCulloch-Pitts neuron
[1943] .
– Inputs are +1 or -1 with real-valued weights.
– If sum of weighted inputs is > 0, then the neuron “fires”
and gives +1 as an output.
– Showed you can comput logical functions.
– Relation to learning proposed (later!) by Donald Hebb
[1949].
• Perceptron model [Rosenblatt, 1958].
– Single-layer network with same kind of neuron.
• Firing when input is about a threshold: ∑xiwi>t .
– Added a learning rule to allow weight selection.
CS-424 Gregory Dudek
Perceptron nets
CS-424 Gregory Dudek
Perceptron learning
• Perceptron learning:
– Have a set of training examples (TS) encoded as input
values (I.e. in the form of binary vectors)
– Have a set of desired output values associated with these
inputs.
• This is supervised learning.
– Problem: how to adjust the weights to make the actual
outputs match the training examples.
• NOTE: we to not allow the topology to change! [You
should be thinking of a question here.]
• Intuition: when a perceptron makes a mistake, it’s weights are
wrong.
– Modify them so make the output bigger or smaller, as desired.
CS-424 Gregory Dudek
Learning algorithm
• Desired Ti
Actual output Oi
• Weight update formula (weight from unit j to i):
Wj,i = Wj,I + k* xj * (Ti - Oi)
Where k is the learning rate.
• If the examples can be learned (encoded), then the
perceptron learning rule will find the weights.
– How?
Gradient descent.
Key thing to prove is the absence of local minima.
CS-424 Gregory Dudek
Perceptrons: what can they learn?
• Only linearly separable functions [Minsky &
Papert 1969].
• N dimensions: N-dimensional hyperplane.
CS-424 Gregory Dudek
More general networks
• Generalize in 3 ways:
– Allow continuous output values [0,1]
– Allow multiple layers.
• This is key to learning a larger class of functions.
– Allow a more complicated function than thresholded
summation
[why??]
Generalize the learning rule to accommodate this: let’s
see how it works.
CS-424 Gregory Dudek
The threshold
• The key variant:
– Change threshold into a differentiable function
– Sigmoid, known as a “soft non-linearity” (silly).
M = ∑xiwi
O = 1 / (1 + e -k M )
CS-424 Gregory Dudek