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CS344: Principles of Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 11, 12: Perceptron Training 30 th and 31 st Jan, 2012

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The Perceptron Model A perceptron is a computing element with input lines having associated weights and the cell having a threshold value. The perceptron model is motivated by the biological neuron. Output = y wnwn W n-1 w1w1 X n-1 x1x1 Threshold = θ

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θ 1 y Step function / Threshold function y = 1 for Σw i x i >=θ =0 otherwise ΣwixiΣwixi

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Features of Perceptron Input output behavior is discontinuous and the derivative does not exist at Σw i x i = θ Σw i x i - θ is the net input denoted as net Referred to as a linear threshold element - linearity because of x appearing with power 1 y= f(net): Relation between y and net is non- linear

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Computation of Boolean functions AND of 2 inputs X1 x2 y 0 0 0 0 10 1 00 1 11 The parameter values (weights & thresholds) need to be found. y w1w1 w2w2 x1x1 x2x2 θ

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Computing parameter values w1 * 0 + w2 * 0 = 0; since y=0 w1 * 0 + w2 * 1 <= θ w2 <= θ; since y=0 w1 * 1 + w2 * 0 <= θ w1 <= θ; since y=0 w1 * 1 + w2 *1 > θ w1 + w2 > θ; since y=1 w1 = w2 = = 0.5 satisfy these inequalities and find parameters to be used for computing AND function.

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Other Boolean functions OR can be computed using values of w1 = w2 = 1 and = 0.5 XOR function gives rise to the following inequalities: w1 * 0 + w2 * 0 = 0 w1 * 0 + w2 * 1 > θ w2 > θ w1 * 1 + w2 * 0 > θ w1 > θ w1 * 1 + w2 *1 <= θ w1 + w2 <= θ No set of parameter values satisfy these inequalities.

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Threshold functions n # Boolean functions (2^2^n) #Threshold Functions (2 n2 ) 1 4 4 2 16 14 3 256 128 4 64K 1008 Functions computable by perceptrons - threshold functions #TF becomes negligibly small for larger values of #BF. For n=2, all functions except XOR and XNOR are computable.

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θ 1 y Step function / Threshold function y = 1 for Σw i x i >=θ =0 otherwise ΣwixiΣwixi

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Features of Perceptron Input output behavior is discontinuous and the derivative does not exist at Σw i x i = θ Σw i x i - θ is the net input denoted as net Referred to as a linear threshold element - linearity because of x appearing with power 1 y= f(net): Relation between y and net is non- linear

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Perceptron Training Algorithm (PTA) Preprocessing: 1. The computation law is modified to y = 1 if ∑w i x i > θ y = o if ∑w i x i < θ ... θ, ≤ w1w1 w2w2 wnwn x1x1 x2x2 x3x3 xnxn... θ, < w1w1 w2w2 w3w3 wnwn x1x1 x2x2 x3x3 xnxn w3w3

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PTA – preprocessing cont… 2. Absorb θ as a weight 3. Negate all the zero-class examples... θ w1w1 w2w2 w3w3 wnwn x2x2 x3x3 xnxn x1x1 w0=θw0=θ x 0 = -1... θ w1w1 w2w2 w3w3 wnwn x2x2 x3x3 xnxn x1x1

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Example to demonstrate preprocessing OR perceptron 1-class,, 0-class Augmented x vectors:- 1-class,, 0-class Negate 0-class:-

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Example to demonstrate preprocessing cont.. Now the vectors are x 0 x 1 x 2 X 1 -1 0 1 X 2 -1 1 0 X 3 -1 1 1 X 4 1 0 0

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Perceptron Training Algorithm 1. Start with a random value of w ex: 2. Test for wx i > 0 If the test succeeds for i=1,2,…n then return w 3. Modify w, w next = w prev + x fail

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PTA on NAND NAND: Y X 2 X 1 Y 0 0 1 0 1 1 W 2 W 1 1 0 1 1 1 0 X 2 X 1 Converted To W 2 W 1 W 0= Θ X 2 X 1 X 0=-1 Θ Θ

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Preprocessing NAND Augmented: NAND-0 class Negated X 2 X 1 X 0 Y X 2 X 1 X 0 0 0 -1 1 V 0: 0 0 -1 0 1 -1 1 V 1: 0 1 -1 1 0 -1 1 V 2: 1 0 -1 1 1 -1 0 V 3: -1 -1 1 Vectors for which W= has to be found such that W. Vi > 0

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PTA Algo steps Algorithm: 1. Initialize and Keep adding the failed vectors until W. Vi > 0 is true. Step 0: W = W 1 = + {V 0 Fails} = W 2 = + {V 3 Fails} = W 3 = + {V 0 Fails} = W 4 = + {V 1 Fails} =

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Trying convergence W 5 = + {V 3 Fails} = W 6 = + {V 1 Fails} = W 7 = + {V 0 Fails} = W 8 = + {V 3 Fails} = W 9 = + {V 2 Fails} =

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Trying convergence W 10 = + {V 3 Fails} = W 11 = + {V 1 Fails} = W 12 = + {V 3 Fails} = W 13 = + {V 1 Fails} = W 14 = + {V 2 Fails} =

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W15 = + {V3 Fails} = W16 = + {V2 Fails} = W17 = + {V3 Fails} = W18 = + {V1 Fails} = W2 = -3, W1 = -2, W0 = Θ = -4 Succeeds for all vectors

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PTA convergence

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Statement of Convergence of PTA Statement: 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.

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