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# 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

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 are independent.

## Boosting

• Weighted Training Set

• Each example has an associated weight $$w_j \geq 0$$.

• The higher the weight, the higher the importance attached to it during the learning of a hypothesis.

• Boosting

• Begin at $$w_j=1$$ for all examples. From this set, it generates the first hypothesis, $$h_1$$.

• Increase the weights for the misclassified examples and decrease the weights for the correctly classified examples to generate $$h_2$$.

• Continue this process until $$K$$ hypothese are generated (where $$K$$ is an input to the algorithm).

• The final ensemble hypothesis is a weighted-majority combination of all $$K$$ hypotheses, each weighted according to how well it performed on the training set.

• If the input learning algorithm $$L$$ is a weak learning algorithm ($$L$$ always return a hypothesis with accuracy that is slightly better than random guessing, $$50\,percent + \epsilon$$.), then ADABOOST will return a hypothesis that classifies the training data perfectly for large enough $$K$$.