Published on January 9, 2008
The delta rule: The delta rule Learn from your mistakes: Learn from your mistakes If it ain’t broke, don’t fix it.: If it ain’t broke, don’t fix it. Outline: Outline Supervised learning problem Delta rule Delta rule as gradient descent Hebb rule Supervised learning: Supervised learning Given examples Find perceptron such that Example: handwritten digits: Example: handwritten digits Find a perceptron that detects “two”s. Delta rule: Delta rule Learning from mistakes. “delta”: difference between desired and actual output. Also called “perceptron learning rule” Two types of mistakes: Two types of mistakes False positive Make w less like x. False negative Make w more like x. The update is always proportional to x. Objective function: Objective function Gradient update Stochastic gradient descent on E=0 means no mistakes. Perceptron convergence theorem: Perceptron convergence theorem Cycle through a set of examples. Suppose a solution with zero error exists. The perceptron learning rule finds a solution in finite time. If examples are nonseparable: If examples are nonseparable The delta rule does not converge. Objective function is not equal to the number of mistakes. No reason to believe that the delta rule minimizes the number of mistakes. Memorization & generalization: Memorization & generalization Prescription: minimize error on the training set of examples What is the error on a test set of examples? Vapnik-Chervonenkis theory assumption: examples are drawn from a probability distribution conditions for generalization contrast with Hebb rule: contrast with Hebb rule Assume that the teacher can drive the perceptron to produce the desired output. What are the objective functions? Is the delta rule biological?: Is the delta rule biological? Actual output: anti-Hebbian Desired output: Hebbian Contrastive Objective function: Objective function Hebb rule distance from inputs Delta rule error in reproducing the output Supervised vs. unsupervised: Supervised vs. unsupervised Classification vs. generation I shall not today attempt further to define the kinds of material [pornography] … but I know it when I see it. Justice Potter Stewart Smooth activation function: Smooth activation function same except for slope of f update is small when the argument of f has large magnitude. Objective function: Objective function Gradient update Stochastic gradient descent on E=0 means zero error. Smooth activation functions are important for generalizing the delta rule to multilayer perceptrons.: Smooth activation functions are important for generalizing the delta rule to multilayer perceptrons.