Artificial Neural NetworksThe future of AIRestaurant Data SetLimited Expressiveness of PerceptronsThe XOR affair • Minsky and Papert (1969) showed certain simple functions cannot be represented (e.g. Boolean XOR). Killed the field! • Mid 80th: Non-linear Neural Networks (Rumelhart et al. 1986)The XOR affairNeural Networks • Rich history, starting in the early forties (McCulloch and Pitts 1943). • Two views: – Modeling the brain – “Just” representation of complex functions (Continuous; contrast decision trees) • Much progress on both fronts. • Drawn interest from: Neuroscience, Cognitive science, AI, Physics, Statistics, and CS/EE.NeuronNeural Structure 1. Cell body; one axon (delivers output to other connect neurons); many dendrites (provide surface area for connections from other neurons). 2. Axon is a single long fiber. 100 or more times the diameter of cell body. Axon connects via synapses to dendrites of other cells. 3. Signals propagated via complicated electrochemical reaction. 4. Each neuron is a “threshold unit”. Neurons do nothing unless the collective influence from all inputs reaches a threshold level. 5. Produces full-strength output. “fires”. Stimulation at some synapses encourages neurons to fire; some discourage from firing. 6. Synapses can increase (excitatory) or decrease (inhibitory) potential (signalWhy Neural Nets? Motivation: Solving problems under the constraints similar to those of the brain may lead to solutions to AI problems that would otherwise be overlooked. • Individual neurons operate very slowly But the brain does complex tasks fast:  massively parallel algorithms • Neurons are failure-prone devices But brain is reliable anyway  distributed representations • Neurons promote approximate matching less brittle  learnableConnectionist Models of Learning Characterized by: • A large number of very simple neuron-like processing elements. • A large number of weighted connections between the elements. • Highly parallel, distributed control. • An emphasis on learning internal representations automatically.Artificial Neurons Activation Functions: stept (x) = 1, if x ≥ t; otherwise 0. sign(x) = +1, if x ≥ 0; otherwise -1 sigmoid(x) = 1/(1+e-x)Example: PerceptronPerceptrons Single Layer Feed Forward Neural Networks Can be easily trained using perceptron algorithm2-Layer Feedforward Networks Boolean functions: • Every boolean function can be represented by network with single hidden layer • But might require exponential (in number of inputs) hidden units Continuous functions: • Every bounded continuous function can be approximated with arbitrarily small error, by network with one hidden layer [Cybenko 1989; Hornik et al. 1989] Any function can be approximated to arbitrary accuracy by a network with two hidden layers [Cybenko 1988]. x1 x2 xN o1 o2 oOMulti-Layer Nets • Fully connected, two layer, feedforward Jonathan Mary Joe Elizabeth Alice Bart How are Mary and Elizabeth related? A=Acquaintances B=Family Activation function: g(x) = (1 if greater than threshold, 0 otherwise)Multi-Layer Nets • Fully connected, two layer, feedforwardOfer Melnik, http://www.demo.cs.brandeis.edu/pr/DIBAOfer Melnik, http://www.demo.cs.brandeis.edu/pr/DIBAOfer Melnik, http://www.demo.cs.brandeis.edu/pr/DIBAHow can we train perceptrons?Hebbian learning • D. O. Hebb: – The general idea is an old one, that any two cells or systems of cells that are repeatedly active at the same time will tend to become 'associated', so that activity in one facilitates activity in the other." (Hebb 1949, p. 70) – "When one cell repeatedly assists in firing another, the axon of the first cell develops synaptic knobs (or enlarges them if they already exist) in contact with the soma of the second cell." (Hebb 1949, p. 63) • Cells that fire together, wire together – If error is small, increase magnitude of connections that contributed. – If error is large, decrease magnitude of connections that contributed.Backpropagation • Classical measure of error – Sum of square errors – hw(x) is output on perceptron on x. • Gradient decent using partial derivatives • Update weightsBackpropagation Training (Overview) Training data: – (x1,y1),…, (xn,yn), with target labels yz {0,1} Optimization Problem (single output neuron): – Variables: network weights wij – Obj.:E=minw∑z=1..n(yz–o(xz))2, – Constraints: none Algorithm: local search via gradient descent. • Randomly initialize weights. • Until performance is satisfactory, – Compute partial derivatives ( E /  wi j) of objective function E for each weight wi j – Update each weight by wi j à wi j +  ( E /  wi j)Smooth and Differentiable Threshold Function • Replace sign function by a differentiable activation function  sigmoid function:Slope of Sigmoid FunctionBackpropagation Training (Detail) • Input: training data (x1,y1),…, (xn,yn), learning rate parameter α. • Initialize weights. • Until performance is satisfactory – For each training instance, • Compute the resulting output • Compute βz = (yz – oz) for nodes in the output layer • Compute βj = ∑k wjk ok (1 – ok) βk for all other nodes. • Compute weight changes for all weights using ∆wi j(l) = oi oj (1 – oj) βj – Add up weight changes for all training instances, and update the weights accordingly. wi,j ← wi,j + α ∑l ∆wi,j(l)Summary: Hidden Units • Hidden units are nodes that are situated between the input nodes and the output nodes. • Hidden units allow a network to learn non-linear functions. • Hidden units allow the network to represent combinations of the input features. • Given too many hidden units, a neural net will simply memorize the input patterns (overfitting). • Given too few hidden units, the network may not be able to represent all of the necessary generalizations (underfitting).How long should you train the net? When would you stop training? A B C D EHow long should you train the net? • The goal is to achieve a balance between correct responses for the training patterns and correct responses for new patterns. – That is, a balance between memorization and generalization) • If you train the net for too long, then you run the risk of overfitting. – Select number of training iterations via cross-validation on a holdout