Announcements 1. Textbook will be on reserve at library 2. Topic schedule change; modified reading assignment: This week: Linear discrimination, evaluating classifiers Extra reading: T. Fawcett, An introduction to ROC analysis, Sections 1-4 (linked from class web page) 3. No class Monday (MLK day) 4. Guest lecture Wednesday: Josh Hugues on multi-layer perceptrons

Perceptrons as simple neural networks w1w1 w2w2 wnwn o w0w0 +1 x1x1 x2x2 xnxn

Geometry of the perceptron Feature 1 Feature 2 Hyperplane In 2d:

Work with one neighbor on this: (a) Find weights (w 0, w 1, w 2 ) for a perceptron that separates “true” and “false” in x 1  x 2. Find the slope and intercept, and sketch the separation line defined by this discriminant, showing that it separates the points correctly. (b) Do the same, but for x 1  x 2. (c) What (if anything) might make one separation line better than another? In-class exercise

Training a perceptron 1.Start with random weights, w = (w 1, w 2,..., w n ). 2.Select training example (x k, t k ). 3.Run the perceptron with input x k and weights w to obtain o. 4.Let  be the learning rate (a user-set parameter). Now, 5.Go to 2.

In-class exercise S = {((0,0), -1), ((0,1), 1), ((1,1), 1)} Let w = {w 0, w 1, w 2 ) = {0.1, 0.1, −0.3} 1. Calculate new perceptron weights after each training example is processed. Let η = What is accuracy on training data after one epoch of training? Did the accuracy improve? o +1 x1x1 0.1 x2x2 −0.3 Perceptron learning rule:

Homework 1 summary w1w1 w2w2 w 64 o w0w0 +1 x1x1 x2x2 x 64 x 1,..., x 64, 8 x 1,..., x 64, 0 Training data: 8 vs Train perceptron: 8 vs Evaluate perceptron : 8 vs. 0 x 1,..., x 64, 8 x 1,..., x 64, 0 Training data: 8 vs Calculate accuracy on training data x 1,..., x 64, 8 x 1,..., x 64, 0 Test data: 8 vs Calculate accuracy on test data Confusion matrix: 8 vs Actual Predicted Give confusion matrix for test data

Homework 1 summary w1w1 w2w2 w 64 o w0w0 +1 x1x1 x2x2 x 64 x 1,..., x 64, 8 x 1,..., x 64, 1 Training data: 8 vs Train perceptron: 8 vs Evaluate perceptron : 8 vs. 1 x 1,..., x 64, 8 x 1,..., x 64, 1 Training data: 8 vs Calculate accuracy on training data x 1,..., x 64, 8 x 1,..., x 64, 1 Test data: 8 vs Calculate accuracy on test data Confusion matrix: 8 vs Actual Predicted Give confusion matrix for test data

Questions on HW What should the “threshold value” be? What should the target and output values look like? The assignment says we will train 10 separate perceptrons; shouldn’t this be 9?

1960s: Rosenblatt proved that the perceptron learning rule converges to correct weights in a finite number of steps, provided the training examples are linearly separable. 1969: Minsky and Papert proved that perceptrons cannot represent non-linearly separable target functions. However, they proved that any transformation can be carried out by adding a fully connected hidden layer.

XOR function x1x1 x2x2 x 1 XOR x x1x1 x2x2

Decision regions of a multilayer feedforward network. The network was trained to recognize 1 of 10 vowel sounds occurring in the context “h_d” (e.g., “had”, “hid”) The network input consists of two parameters, F1 and F2, obtained from a spectral analysis of the sound. The 10 network outputs correspond to the 10 possible vowel sounds. (From T. M. Mitchell, Machine Learning ) Multi-layer perceptron example

Good news: Adding hidden layer allows more target functions to be represented. Bad news: No algorithm for learning in multi-layered networks, and no convergence theorem! Quote from Minsky and Papert’s book, Perceptrons (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.”

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? Good news: Successful credit-apportionment learning algorithms developed soon afterwards (e.g., back- propagation). Still successful, in spite of lack of convergence theorem.