CAP6938Neuroevolution and Artificial Embryogeny

Neural Network Weight Optimization

Dr. Kenneth StanleyJanuary 18, 2006

Review

• Remember, the values of the weights and the topology determine the functionality

• Given a topology, how are weights optimized?• Weights are just parameters on a structure

? ??

??

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Two Cases

• Output targets are known• Output targets are not known

X1 X2

H1 H2

out1 out2

w11

w21w12

Decision Boundaries

++

- +

OR function:

1 1 1 1 -1 1 -1 1 1-1 -1 -1

Input Output

• OR is linearly separable• Linearly separable problems do not

require hidden nodes (nonlinearities)

Bias

Decision Boundaries

• XOR is not linearly separable• Requires at least one hidden node

-+

- +

XOR function:

1 1 -1 1 -1 1 -1 1 1-1 -1 -1

Input Output

Bias

Hebbian Learning

• Change weights based on correlation of connected neurons

• Learning rules are local• Simple Hebb Rule: • Works best when relevance of inputs to

outputs is independent• Simple Hebb Rule grows weights unbounded• Can be made incremental:

yxww iii )old(new)(

yxw ii

More Complex Local Learning Rules

• Hebbian Learning with a maximum magnitude:– Excitatory: – Inhibitory:

• Second terms are decay terms: forgetting– Happens when presynaptic node does not affect

postsynaptic node• Other rules are possible• Videos: watch the connections change

)01 21 . Wx(yη(W-w)xyw )0.1( 21 y(W-w)x(W-w)xyw

Perceptron Learning

• Will converge on correct weights• Single layer learning rule:• Rule is applied until boundary is learned

Bias

iii txww )old(new)(

Backpropagation• Designed for at least one hidden layer• First, activation propagates to outputs• Then, errors are computed and assigned• Finally, weights are updated• Sigmoid is a common activation function

X1 X2

z1 z2

y1 y2

v11v21v12

v22

w11w21w12

w22

t1 t2 x’s are inputs

z’s are hidden units

y’s are outputs

t’s are targets

v’s are layer 1 weights

w’s are layer 2 weights

Backpropagation Algorithm1) Initialize weights2) While stopping condition is false, for each training pair

1) Compute outputs by forward activation2) Backpropagate error:

1) For each output unit, error2) Weight correction3) Send error back to hidden units4) Calculate error contribution for each hidden unit:

5) Weight correction

3) Adjust weights by adding weight corrections

)()( kkkk yinfyt (target minus output times slope)

jkjk zw (Learning rate times error times hidden output)

)(1

j

m

kjkkj zinfw

ijij xv

Example Applications

• Anything with a set of examples and known targets

• XOR• Character recognition• NETtalk: reading English aloud• Failure predicition• Disadvantages: trapped in local optima

Output Targets Often Not Available

(Stone, Sutton, and Kuhlmann 2005)

One Approach: Value Function Reinforcement Learning

• Divide the world into states and actions• Assign values to states• Gradually learn the most promising states

and actions

Start

Goal0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Learning to Navigate

Start

Goal0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0Start

Goal0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0.5

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Start

Goal0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0.9 1Start

Goal0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 1

1 1 1 1 1 1 1 1

T=1 T=56

T=350 T=703

How to Update State/Action Values

• Q learning rule:

• Exploration increases Q-values’ accuracy• The best actions to take in different states

become known• Works only in Markovian domains

),(),(),( allactionsnextstateQMaxactionstateRactionstateQ

Backprop In RL

• The state/action table can be estimated by a neural network

• The target learned by the network is the Q-value:

NNAction State_description

Value

Next Week: Evolutionary Computation

For 1/23: Mitchell ch.1 (pp. 1-31) and ch.2 (pp. 35-80) Note Section 2.3 is "Evolving Neural Networks"For 1/25: Mitchell pp. 117-38, paper: No Free Lunch Theorems for Optimization (1996) by David H. Wolpert, William G. Macready

• EC does not require targets• EC can be a kind of RL• EC is policy search• EC is more than RL