NEURAL NETWORKS CHAPTER 20,SECTION 5 Chapter 20.Section 5 1
Neural networks Chapter 20, Section 5 Chapter 20, Section 5 1
Outline ◇Brains ◇ Neural networks ◇Perceptrons Multilayer perceptrons Applications of neural networks Chapter 20,Section 5 2
Outline ♦ Brains ♦ Neural networks ♦ Perceptrons ♦ Multilayer perceptrons ♦ Applications of neural networks Chapter 20, Section 5 2
Brains 1011 neurons of >20 types,1014 synapses,1ms-10ms cycle time Signals are noisy "spike trains"of electrical potential Axonal arborization Q Axon from another cell Synapse Dendrite Axon Nucleus Synapses Cell body or Soma Chapter 20.Section 5 3
Brains 1011 neurons of > 20 types, 1014 synapses, 1ms–10ms cycle time Signals are noisy “spike trains” of electrical potential Axon Cell body or Soma Nucleus Dendrite Synapses Axonal arborization Axon from another cell Synapse Chapter 20, Section 5 3
McCulloch-Pitts "unit" Output is a "squashed"linear function of the inputs: ai-g(in)=g(②jW.a) Bias Weight a0=-1 Wo.i ai=g(inj) 形a In, a Input Input Output Links Activation Output Function Function Links A gross oversimplification of real neurons,but its purpose is to develop understanding of what networks of simple units can do Chapter 20,Section 5 4
McCulloch–Pitts “unit” Output is a “squashed” linear function of the inputs: ai ← g(ini) = g ΣjWj,iaj Output Σ Input Links Activation Function Input Function Output Links a0 = −1 ai = g(ini) ai g W ini j,i W0,i Bias Weight aj A gross oversimplification of real neurons, but its purpose is to develop understanding of what networks of simple units can do Chapter 20, Section 5 4
Activation functions g(in) g(ini) +1 ini (a) (b) (a)is a step function or threshold function (b)is a sigmoid function 1/(1+e-*) Changing the bias weight Wo.:moves the threshold location Chapter 20.Section 55
Activation functions (a) (b) +1 +1 ini ini g(ini g(in ) i) (a) is a step function or threshold function (b) is a sigmoid function 1/(1 + e−x) Changing the bias weight W0,i moves the threshold location Chapter 20, Section 5 5
Implementing logical functions Wo=1.5 Wo=0.5 W0=-0.5 = W1=-1 W,=1 W2=1 AND OR NOT McCulloch and Pitts:every Boolean function can be implemented Chapter 20,Section 5 6
Implementing logical functions AND W0 = 1.5 W1 = 1 W2 = 1 OR W2 = 1 W1 = 1 W0 = 0.5 NOT W1 = –1 W0 = – 0.5 McCulloch and Pitts: every Boolean function can be implemented Chapter 20, Section 5 6
Network structures Feed-forward networks: single-layer perceptrons multi-layer perceptrons Feed-forward networks implement functions,have no internal state Recurrent networks: -Hopfield networks have symmetric weights(Wij-Wj) g(x)=sign(x),ai=+1;holographic associative memory Boltzmann machines use stochastic activation functions, ≈MCMC in Bayes nets recurrent neural nets have directed cycles with delays have internal state (like flip-flops),can oscillate etc. Chapter 20.Section 5 7
Network structures Feed-forward networks: – single-layer perceptrons – multi-layer perceptrons Feed-forward networks implement functions, have no internal state Recurrent networks: – Hopfield networks have symmetric weights (Wi,j = Wj,i) g(x) = sign(x), ai = ± 1; holographic associative memory – Boltzmann machines use stochastic activation functions, ≈ MCMC in Bayes nets – recurrent neural nets have directed cycles with delays ⇒ have internal state (like flip-flops), can oscillate etc. Chapter 20, Section 5 7
Feed-forward example 3 W* W35 5 W23 W 4 W45 Feed-forward network=a parameterized family of nonlinear functions: a5=g(W3,5·a3+W4,5·a4) =g(W3,5·g(W,3·a1+W2,3·a2)+W4,5·g(W,4·a1+W2,4·a2) Adjusting weights changes the function:do learning this way! Chapter 20,Section 5 8
Feed-forward example W1,3 W1,4 W2,3 W2,4 W3,5 W4,5 1 2 3 4 5 Feed-forward network = a parameterized family of nonlinear functions: a5 = g(W3,5 · a3 + W4,5 · a4) = g(W3,5 · g(W1,3 · a1 + W2,3 · a2) + W4,5 · g(W1,4 · a1 + W2,4 · a2)) Adjusting weights changes the function: do learning this way! Chapter 20, Section 5 8
Single-layer perceptrons Perceptron output 0.8 0.6 0.4 0.2 0 、0 Input Output Units Units Output units all operate separately-no shared weights Adjusting weights moves the location,orientation,and steepness of cliff Chapter 20.Section 5 9
Single-layer perceptrons Input Units Units Output Wj,i -4 -2 0 2 x1 4 -4 -2 0 2 4 x2 0 0.2 0.4 0.6 0.8 1 Perceptron output Output units all operate separately—no shared weights Adjusting weights moves the location, orientation, and steepness of cliff Chapter 20, Section 5 9
Expressiveness of perceptrons Consider a perceptron with g=step function(Rosenblatt,1957,1960) Can represent AND,OR,NOT,majority,etc.,but not XOR Represents a linear separator in input space: ∑，Wz)>0orW·x>0 0( 00 0 0 (a)xi and x2 (b)x1 orx2 (c)x1 xor x2 Minsky Papert (1969)pricked the neural network balloon Chapter 20,Section 5 10
Expressiveness of perceptrons Consider a perceptron with g = step function (Rosenblatt, 1957, 1960) Can represent AND, OR, NOT, majority, etc., but not XOR Represents a linear separator in input space: ΣjWjxj > 0 or W · x > 0 (a) x1 and x2 1 0 0 1 x1 x2 (b) x1 or x2 0 1 1 0 x1 x2 (c) x1 xor x2 ? 0 1 1 0 x1 x2 Minsky & Papert (1969) pricked the neural network balloon Chapter 20, Section 5 10
