CS344: Principles of Artificial Intelligence. Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 13,14: Perceptron Training 2 nd Feb, 2012. Perceptron Training Algorithm. Start with a random value of w ex: <0,0,0…> Test for wx i > 0 If the test succeeds for i=1,2,…n
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Pushpak BhattacharyyaCSE Dept., IIT Bombay
Lecture 13,14: Perceptron Training
2nd Feb, 2012
If the test succeeds for i=1,2,…n
then return w
3. Modify w, wnext = wprev + xfail
Whatever be the initial choice of weights and whatever be the vector chosen for testing, PTA converges if the vectors are from a linearly separable function.
wi+1 = wi + Xj
w* s.t. w*Xj > 0 j
G(wn) = wn . w*
where wn = weight at nth iteration
where = angle between wn and w*
wn . w* = (wn-1 + Xn-1fail ) . w*
w*.Xifail is always positive: note carefully
=> Numerator grows faster than denominator
. . .
xnConcept of Hyper-planes
defines a perceptron.
w.x>0 gives y=1
w.x≤0 gives y=0
Depending on different values of
w and θ, four different functions are possible
1Simple perceptron contd.
Substituting x=0 and x=1,
we get θ=0 and w=θ.
These two lines intersect to
form four regions, which
correspond to the four functions.
i.e. Rm,d = ?
We work in the <X1, X2> space or the <w1, w2, Ѳ> space
W1 = W2 = 1, Ѳ = 0.5
X1 + x2 = 0.5
General equation of a Hyperplane:
Σ Wi Xi = Ѳ
(Line in 2-D)
Regions produced by lines not necessarily passing through origin
L2: 2+2 = 4
L2: 2+2+3 = 7
L2: 2+2+3+4 = 11
New regions created = Number of intersections on the incoming line by the original lines
Total number of regions = Original number of regions + New regions created
P1, P2, P3 and P4 are planes in the <W1,W2, Ѳ> space
Number of regions = 2+2 = 4
Number of regions = 4 + 4 = 8
Number of regions = 8 + 6 = 14
W1*X1 + W2*X2 > Ѳ
W1’*X1 + W2’*X2 > Ѳ’
If <W1,W2, Ѳ> and <W1’,W2’, Ѳ’> share a region then they compute the same function