EECS 492 Chapter 18: Learning From Examples

Introduction

  • An agent is learning if it improves its performance on future tasks after making observations about the world.

  • This chapter: From a collection of input-output pairs, learn a function that predicts the output for new inputs.

Supervised Learning

  • Task
    Given a training set of N example input-output pairs.
    \((x_1, y_1), (x_2, y_2), ... (x_N, y_N)\)
    where each \(y_j\) was generated by an unkown function \(y = f(x)\), discover a function \(h\) that approximates the true function \(f\).

  • Function \(h\): Hypothesis.

    • Learning is a search through space of possible hypotheses for one that performs well, even on new examples.

    • To measure accuracy of a hypothesis, we use a test set of examples distinct from the training set.

    • A hypothesis generalizes if it correctly predits the value of \(y\) for novel examples.

    • Sometimes \(f\) is stochastic-not strictly a function of \(x\)- which means that we have to learn a conditional probability distribution \(P(Y|x)\)

  • Types of Learning Problems

    • Classification: Type of learning problem for which the output \(y\) is one of a finite set of values (such as \(sunny\), \(cloudy\), or \(rainy\)).

    • Regression: Type of learning problem for which the output \(y\) is a number (such as temperature).

  • Ockham’s Razor: Choose the simplest hypothesis consistent with the data.

  • “In general, there is a tradeoff between complex hypotheses that fit the training data well and simpler hypotheses that may generalize better.”

  • Supervised learning is done by choosing the hypothesis \(h^*\) that is most probably given the data:
    \(h^*=argmax_{h\in H}\,P(h|data).\)
    By Bayes: \(h^*=argmax_{h\in H}\,P(data|h)P(h).\)
    \(P(h)\) is high for a degree 1/2 polynomial and low for a higher degree polynomial.

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Ensemble Learning 18.10 p748

  • So far: Looked at learning methods in which a single hypothesis, chosen from a hypothesis space, is used to make predictions

  • Ensemble Learning: Select a collection, or ensemble, of hypotheses from the hypothesis space and combine their predictions.

  • Motivation: Consider we have an ensemble of K=5 hypotheses and suppose we combine their predictions using simple majority voting. For the ensemble to misclassify a new example, at least three of the five hypotheses have to misclassify it. That is, there is a lower chance of misclassification than with a single hypothesis.

    • Suppose each hypothesis \(h_k\) in the ensemble has an error of \(p\) (the probability that a randomly chosen example is misclassified by \(h_k\) is \(p\).

    • Suppose that the errors made by each hypothesis a