1. Find a hypothesis class H and a sequence of examples on which Consistent makes | H |- 1 mistakes.

1. Find a
hypothesis class H and a sequence of examples on which Consistent makes |H|_1 mistakes.

2. Find a
hypothesis class H and a sequence of examples on which the mistake bound
the Halving algorithm is tight.

3. Let d _ 2, X = {1,
. . .,d} and
let H = {h j : j _ [d]}, where h j
(x) = 1[x=j ]. Calculate MHalving(H) (i.e.,
derive lower and upper bounds on MHalving(H), and prove that they
are equal).

4 The Doubling
Trick:

In Theorem 21.15, the parameter _ depends
on the time horizon T . In this exercise we show how to get rid of this dependence
by a simple trick. Consider an
View complete question

1. Find a
hypothesis class H and a sequence of examples on which Consistent makes |H|_1 mistakes.

2. Find a
hypothesis class H and a sequence of examples on which the mistake bound
the Halving algorithm is tight.

3. Let d _ 2, X = {1,
. . .,d} and
let H = {h j : j _ [d]}, where h j
(x) = 1[x=j ]. Calculate MHalving(H) (i.e.,
derive lower and upper bounds on MHalving(H), and prove that they
are equal).

4 The Doubling
Trick:

In Theorem 21.15, the parameter _ depends
on the time horizon T . In this exercise we show how to get rid of this dependence
by a simple trick. Consider an algorithm that enjoys a regret bound of the form
_ _ T , but its

parameters
require the knowledge of T . The doubling trick, described in the
following, enables us to convert such an algorithm into an algorithm that does
not need to know the time horizon. The idea is to divide the time into periods
of increasing size and run the original algorithm on each period. The Doubling
Trick

input: algorithm A whose parameters depend on
the time horizon for m = 0,1,2,
. . . run A on the 2m
rounds t = 2m, . . .,2m+1 _1 Show that if the regret of A on each period of 2m rounds
is at most _ _ 2m, then the total regret is at most

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