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Neural Networks. Examples of Single Layer Perceptron Convergence in Linear Case No Convergence in Non-linear Case (Hyper)Line of separation Separation of space by (hyper)line/plane Orthogonality to weight vector. Neural Networks.

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neural networks
Neural Networks
  • Examples of Single Layer Perceptron
    • Convergence in Linear Case
    • No Convergence in Non-linear Case
  • (Hyper)Line of separation
    • Separation of space by (hyper)line/plane
    • Orthogonality to weight vector

Lecture 7, CS567

neural networks1
Neural Networks
  • Consider 2 input single neuron for Boolean prediction of clinical drug effectiveness, based on results from 2 experiments X and Y
    • Neuron output = 1 => Clinically effective drug
    • Neuron output = 0 => Drug not effective clinically
  • Inputs to network
    • p1 = Prediction of Experiment X [Yes/No] = [0/1]
    • p2 = Prediction of Experiment Y [Yes/No] = [0/1]
  • Consider wonder drug candidates dA, dB, dC, dD for revolutionary product “eMemory” that is proclaimed to increase IQ (as it turns on genes responsible for mental alertness and suppression of neuronal loss) used to train network

Lecture 7, CS567

neural networks2
Neural Networks

0 = No effect; 1 = Effect observed

Lecture 7, CS567

neural networks3
Neural Networks
  • Consider 2 input SLP with random initial weights and bias

p1

w1,1=1

b=0.5

Sum = x = w1,1p1 + w1,2p2 + b

= p1 + p2 + 0.5

Neuron

w1,2=1

p2

Output = fhardlimit(x) = 0 if (x < 0), else 1

Lecture 7, CS567

neural networks4
Neural Networks
  • Consider NN prediction for dA (Target prediction is 0)
  • Note that output now for dA = -0.5 => 0 (as expected)

p1

0

w1,1=1

b=0.5

Sum = 0.5; Output f(0.5)= 1

Neuron

w1,2=1

p2

0

Error e = Target - f(x) = 0 – 1 = -1

Wcurrent = Wprevious + eP => w1,1=1; w1,2=1

bcurrent = bprevious + e => b = 0.5 – 1 = -0.5

Lecture 7, CS567

neural networks5
Neural Networks
  • Consider NN prediction for dB (Target prediction is 0)
  • Note that output now for dB = -0.5 => 0 (as expected)

p1

1

w1,1=1

b=-0.5

Sum = 0.5; Output f(0.5)= 1

Neuron

w1,2=1

p2

0

Error e = Target - f(x) = 0 – 1 = -1

Wcurrent = Wprevious + eP => w1,1=0; w1,2=1

bcurrent = bprevious + e => b = -0.5 – 1 = -1.5

Lecture 7, CS567

neural networks6
Neural Networks
  • Consider NN prediction for dC (Target prediction is 0)

p1

0

w1,1=0

b=-1.5

Sum = -0.5; Output f(-0.5)= 0

Neuron

w1,2=1

p2

1

Error e = Target - f(x) = 0 – 0 = 0

No change in weights or bias

Lecture 7, CS567

neural networks7
Neural Networks
  • Consider NN prediction for dD (Target prediction is 1)
  • Note that output now for dD = 1.5 => 1 (as expected)

p1

1

w1,1=0

b=-1.5

Sum = -0.5; Output f(-0.5)= 0

Neuron

w1,2=1

p2

1

Error e = Target - f(x) = 1 - 0 = 1

Wcurrent = Wprevious + eP => w1,1=1; w1,2=2

bcurrent = bprevious + e => b = -1.5 + 1 = -0.5

Lecture 7, CS567

neural networks8
Neural Networks
  • Check NN prediction is correct for all inputs dA:dD
  • OK for all except dC (expect 0)

p1

0

w1,1=1

b=-0.5

Sum = 1.5; Output f(1.5)= 1

Neuron

w1,2=2

p2

1

Error e = Target - f(x) = 0 - 1 = -1

Wcurrent = Wprevious + eP => w1,1=1; w1,2=1

bcurrent = bprevious + e => b = -0.5 - 1 = -1.5

Lecture 7, CS567

neural networks9
Neural Networks
  • Check NN prediction is correct for all inputs dA:dD
  • Converged: Gives right prediction for all possible inputs. (Try starting with different initial states)

p1

w1,1=1

b=-1.5

Neuron

w1,2=1

p2

Lecture 7, CS567

neural networks hyper line of separation
Neural Networks- (Hyper)Line of separation
  • 2 input Boolean problem is linear if line can be drawn between the two classes of result
  • 3 input Boolean problem is linear if plane can be drawn between the two classes of result
  • Note that Weight vector points to the positive side and is orthogonal to line of separation

p2

2

Effective drug

W

1

Ineffective drug

p1

1

2

WP + b = 0

Lecture 7, CS567

neural networks10
Neural Networks
  • Will this converge? (XOR does not converge)

Lecture 7, CS567