Carla P. Gomes CS4700 CS 4700: Foundations of Artificial Intelligence Prof. Carla P. Gomes Module: Neural Networks Expressiveness of Perceptrons (Reading: Chapter 20.5)

Carla P. Gomes CS4700 Expressiveness of Perceptrons

Carla P. Gomes CS4700 Expressiveness of Perceptrons What hypothesis space can a perceptron represent? Even more complex Booelan functions such as majority function. But can it represent any arbitrary Boolean function?

Carla P. Gomes CS4700 Expressiveness of Perceptrons A threshold perceptron returns 1 iff the weighted sum of its inputs (including the bias) is positive, i.e.,: I.e., iff the input is on one side of the hyperplane it defines. Linear discriminant function or linear decision surface. Weights determine slope and bias determines offset. Perceptron  Linear Separator

Carla P. Gomes CS4700 x1x1 x2x Can view trained network as defining a “separation line”. Linear Separability Percepton used for classification Consider example with two inputs, x1, x2: What is its equation?

Carla P. Gomes CS4700 Linear Separability x1x1 x2x2   OR

Carla P. Gomes CS4700 Linear Separability x1x1 x2x2   AND

Carla P. Gomes CS4700 Linear Separability x1x1 x2x2   XOR

Carla P. Gomes CS4700 Linear Separability x1x1 x2x2   XOR Minsky & Papert (1969) Bad News: Perceptrons can only represent linearly separable functions. Not linearly separable

Carla P. Gomes CS4700 Consider a threshold perceptron for the logical XOR function (two inputs): Our examples are: x1x2label Linear Separability: XOR Given our examples, we have the following inequalities for the perceptron: From (1) ≤ T  T  0 From (2) w > T  w 1 > T From (3) 0 + w2 > T  w 2 > T From (4) w1 + w2 ≤ T w1 + w2 > 2T contradiction So, XOR is not linearly separable

Carla P. Gomes CS4700 Convergence of Perceptron Learning Algorithm … training data linearly separable … step size  sufficiently small … no “hidden” units Perceptron converges to a consistent function, if…

Perceptron learns majority function easily, DTL is hopeless

DTL learns restaurant function easily, perceptron cannot represent it

Carla P. Gomes CS4700 Good news: Adding hidden layer allows more target functions to be represented. Minsky & Papert (1969)

Carla P. Gomes CS4700 Multi-layer Perceptrons (MLPs) Single-layer perceptrons can only represent linear decision surfaces. Multi-layer perceptrons can represent non-linear decision surfaces.

Carla P. Gomes CS4700 Minsky & Papert (1969) “[The perceptron] has many features to attract attention: its linearity; its intriguing learning theorem; its clear paradigmatic simplicity as a kind of parallel computation. There is no reason to suppose that any of these virtues carry over to the many-layered version. Nevertheless, we consider it to be an important research problem to elucidate (or reject) our intuitive judgment that the extension is sterile.” Bad news: No algorithm for learning in multi-layered networks, and no convergence theorem was known in 1969! Minsky & Papert (1969) pricked the neural network balloon …they almost killed the field. Winter of Neural Networks Rumors say these results may have killed Rosenblatt….

Carla P. Gomes CS4700 Two major problems they saw were 1.How can the learning algorithm apportion credit (or blame) to individual weights for incorrect classifications depending on a (sometimes) large number of weights? 2.How can such a network learn useful higher-order features?

Carla P. Gomes CS4700 Good news: Successful credit-apportionment learning algorithms developed soon afterwards (e.g., back-propagation). Still successful, in spite of lack of convergence theorem. The “Bible” (1986)