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….

## Show that E[LD(hr )] ≤ MA(H) T , where the expectation is over the random choice of the instances as well as a random choice of r according to the uniform distribution over [T ].

1. Online-to-batch Conversions: In this exercise we demonstrate how a successful online learning algorithm can be used to derive a successful PAC learner as well. Consider a PAC learning problem for binary classification parameterized by an instance domain, *X*, and a hypothesis class, *H*. Suppose that there exists an online learning algorithm, *A*, which enjoys a mistake bound *MA*(*H*) ∞. Consider running this algorithm on a sequence of *T *examples which are sampled i.i.d. from a distribution *D *over the instance space *X*, and are labeled by some *h**_ *∈ *H*. Suppose that for every round *t*, the prediction of the algorithm is based on a hypothesis *ht *: *X *→{0*,*1}. Show that E[*LD*(*hr *)] ≤ *MA*(*H*) *T **,* where the expectation is over the random choice of the instances as well as a random choice of *r *according to the uniform distribution over [*T *].

*Hint: *Use similar arguments to the ones appearing in the proof of Theorem 14.8.