Learning with Neural Networks. Artificial Intelligence CMSC 25000 February 19, 2002. Agenda. Neural Networks: Biological analogy Review: singlelayer perceptrons Perceptron: Pros & Cons Neural Networks: Multilayer perceptrons Neural net training: Backpropagation Strengths & Limitations
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Dendrites
Axon
Nucleus
Cell Body
Neurons: Receive inputs from other neurons (via synapses)
When input exceeds threshold, “fires”
Sends output along axon to other neurons
Brain: 10^11 neurons, 10^16 synapses
Single neuronlike element
Binary inputs &output
Weighted sum of inputs > threshold
y
w0
wn
w1
w3
w2
x0=1
x1
x2
x3
xn
. . .
compensates for threshold
x0 w0
x2
0
0
0
0
+ +++ + +
0
0
0
x1
Perceptron Learningx2
+
0
But not
0
+
x1
xor
X1
Y1
Y2
X2
X3
X4
Inputs
Hidden
Hidden
Outputs
o1
w11
Network
Topology:
2 hidden nodes
1 output
w13
x1
w01
w21
y
1
w23
w12
w03
w22
x2
1
w02
o2
Desired behavior:
x1 x2 o1 o2 y
0 0 0 0 0
1 0 0 1 1
0 1 0 1 1
1 1 1 1 0
1
Weights:
w11= w12=1
w21=w22 = 1
w01=3/2; w02=1/2; w03=1/2
w13=1; w23=1
z1
z2
z3
y3
z3
w03
1
w23
w13
y1
y2
z2
z1
w21
w01
w22
w02
w11
1
w12
1
x2
x1
Neural Net Examplexi : ith sample input vector
w : weight vector
yi*: desired output for ith sample

Sum of squares error over training samples
From 6.034 notes lozanoperez
Full expression of output in terms of input and weights
Error as function of weights
Find rate of change of error
Follow steepest rate of change
Change weights s.t. error is minimized
Gradient DescentdG
dw
E
G(w)
w0w1
w
Local
minima
z1
z2
z3
y3
z3
w03
1
w23
w13
y1
y2
z2
z1
w21
w01
w22
w02
w11
1
w12
1
x2
x1
Gradient of Error
Note: Derivative of sigmoid:
ds(z1) = s(z1)(1s(z1))
dz1
From 6.034 notes lozanoperez
MIT AI lecture notes, LozanoPerez 2000
i
j
k
y3
z3
w03
1
w13
y1
w23
y2
z2
z1
w21
w01
w22
w02
1
w11
w12
1
x2
x1
Backprop ExampleForward prop: Compute zi and yi given xk, wl
From 6.034 notes lozanoperez