1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## Prove that Memorize is a consistent learner for every class of (binary-valued) functions over any countable domain.

1. Prove that Memorize is a consistent learner for every class of (binary-valued) functions over any countable domain.

2. Let *H *be the class of intervals on the line (formally equivalent to axis aligned rectangles in dimension *n *= 1). Propose an implementation of the ERM*H *learning rule (in the agnostic case) that given a training set of size *m*, runs in time *O*(*m*2). *Hint*: Use dynamic programming.

3. Let *H*1*,**H*2*, . . . *be a sequence of hypothesis classes for binary classification. Assume that there is a learning algorithm that implements the ERM rule in the realizable case such that the output hypothesis of the algorithm for each class *Hn *only depends on *O*(*n*) examples out of the training set. Furthermore, assume that such a hypothesis can be calculated given these *O*(*n*) examples in time *O*(*n*), and that the empirical risk of each such hypothesis can be evaluated in time *O*(*mn*). For example, if *Hn *is the class of axis aligned rectangles in R*n*, we saw that it is possible to find an ERM hypothesis in the realizable case that is defined by at most 2*n *examples. Prove that in such cases, it is possible to find an ERMhypothesis for*Hn *in the unrealizable case in time *O*(*mnmO*(*n*)).