# 0.9 Error bounds in countably infinite spaces

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## Introduction

In the last lecture , we studied bounds of the following form: for any $\delta >0$ , with probability at least $1-\delta$ ,

$R\left(f\right)\le {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{log|\mathcal{F}|+log\left(\frac{1}{\delta }\right)}{2n}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall f\in \mathcal{F}$

which led to upper bounds on the estimation error of the form

$E\left[R\left({\stackrel{^}{f}}_{n}\right)\right]-\underset{f\in F}{min}R\left(f\right)\phantom{\rule{4pt}{0ex}}\le \phantom{\rule{4pt}{0ex}}\sqrt{\frac{log|\mathcal{F}|+log\left(n\right)+2}{n}}.$

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

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}<\infty .$

The numbers c( $f$ ) can be interpreted as

• measures of complexity
• -log of prior probabilities
• codelengths

In particular, if P( $f$ ) is the prior probability of $f$ then

${e}^{-\left(-,log,p,\left(,f,\right)\right)}=p\left(f\right)$

so $c\left(f\right)\equiv -logp\left(f\right)$ produces

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}=\sum _{f\in \mathcal{F}}p\left(f\right)=1.$

Now recall Hoeffding's inequality. For each $f$ and every $ϵ>0$

$P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,ϵ\right)\le {e}^{-2n{ϵ}^{2}}$

or for every $\delta >0$

$P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta }\right)}{2n}}\right)\le \delta .$

Suppose $\delta >0$ is specified. Using the values c( $f$ ) for $f\in \mathcal{F}$ , define

$\delta \left(f\right)={e}^{-c\left(f\right)}\delta .$

Then we have

$P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\le \delta \left(f\right).$

Furthermore we can apply the union bound as follows

$\begin{array}{ccc}\hfill P\left(\underset{f\in \mathcal{F}}{sup},\left\{R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),-,\sqrt{\frac{log\left(1/\delta \left(f\right)\right)}{2n}}\right\},\ge ,0\right)& \le & P\left(\bigcup _{f\in \mathcal{F}},R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\hfill \\ & \le & \sum _{f\in \mathcal{F}}P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\hfill \\ & \le & \sum _{f\in \mathcal{F}}\delta \left(f\right)=\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}\delta =\delta \hfill \end{array}.$

So for any $\delta >0$ with probability at least $1-\delta$ , we have that $\forall f\in \mathcal{F}$

$\begin{array}{ccc}\hfill R\left(f\right)& \le & {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\hfill \\ & =& {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{c\left(f\right)+log\left(\frac{1}{\delta }\right)}{2n}}\hfill \end{array}.$

## Special case

Suppose $\mathcal{F}$ is finite and $c\left(f\right)=log|\mathcal{F}|\phantom{\rule{1.em}{0ex}}\forall f\in \mathcal{F}$ . Then

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}=\sum _{f\in \mathcal{F}}{e}^{-log|\mathcal{F}|}=\sum _{f\in \mathcal{F}}\frac{1}{|\mathcal{F}|}=1$

and

$\delta \left(f\right)=\frac{\delta }{|\mathcal{F}|}$

which implies that for any $\delta >0$ with probability at least $1-\delta$ , we have

$R\left(f\right)\le {\stackrel{^}{R}}_{n}\left(f\right)+\sqrt{\frac{log|\mathcal{F}|+log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall f\in \mathcal{F}.$

Note that this is precisely the bound we derived in the last lecture .

Choosing c( $f$ )

The generalized bounds allow us to handle countably infinite collections of candidate functions, but we require that

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}<\infty .$

Of course, if $c\left(f\right)=-logp\left(f\right)$ where $p\left(f\right)$ is a proper prior probability distribution then we have

$\sum _{f\in \mathcal{F}}{e}^{-c\left(f\right)}=1.$

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

$\begin{array}{ccc}\hfill P\left(\bigcup _{f\in \mathcal{F}},R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)& \le & \sum _{f\in \mathcal{F}}P\left(R,\left(f\right),-,{\stackrel{^}{R}}_{n},\left(f\right),\ge ,\sqrt{\frac{log\left(\frac{1}{\delta \left(f\right)}\right)}{2n}}\right)\hfill \\ & \le & \sum _{f\in \mathcal{F}}\delta \left(f\right)=\sum _{f\in \mathcal{F}}{2}^{-c\left(f\right)}\delta =\delta \hfill \end{array}.$

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