The key assumptions made
in deriving the error bounds were:
bounded loss function
finite collection of candidate functions
The bounds are valid for every
${P}_{XY}$ and are called
distribution-free.
Deriving bounds for countably infinite spaces
In this lecture we will generalize the previous results in a
powerful way by developing bounds applicable to possibly infinitecollections of candidates. To start let us suppose that
$\mathcal{F}$ is a
countable, possibly infinite, collection of candidate functions.Assign a positive number c(
$f$ ) to each
$f\in \mathcal{F}$ , such that
However, it may be difficult to design a probability distribution
over an infinite class of candidates. The coding perspectiveprovides a very practical means to this end.
Assume that we have assigned a uniquely decodable binary code to
each
$f\in \mathcal{F}$ , and let c(
$f$ ) denote the codelength for
$f$ . That
is, the code for
$f$ is c(
$f$ ) bits long. A very useful class of
uniquely decodable codes are called prefix codes.
Prefix Code
A code is called a prefix codeif no codeword is a
prefix of any other codeword.
The kraft inequality
For any binary prefix code, the codeword lengths
${c}_{1}$ ,
${c}_{2}$ , ...
satisfy
$$\sum _{i=1}^{\infty}{2}^{-{c}_{i}}\le 1.$$
Conversely, given any
${c}_{1}$ ,
${c}_{2}$ , ... satisfying the inequality
above we can construct a prefix code with these codeword lengths.We will prove this result a bit later, but now let's see how this
is useful in our learning problem.
Assume that we have assigned a binary prefix codeword to each
$f\in \mathcal{F}$ , and let c(
$f$ ) denote the bit-length of the codeword for
$f$ . Set
$\delta \left(f\right)={2}^{-c\left(f\right)}\delta $ . Then
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