# 0.3 Introduction to complexity regularization  (Page 3/3)

 Page 3 / 3

The number of unique labellings of the training data that can be achieved with linear classifiers is, in fact, finite. A line can bedefined by picking any pair of training points, as illustrated in [link] . Two classifiers can be defined from each such line: one that outputs a label “1” for everything on or abovethe line, and another that outputs “0” for everything on or above. There exist $\left(\genfrac{}{}{0pt}{}{n}{2}\right)$ such pairs of training points, and these define all possible unique labellings of the training data.Therefore, there are at most $2\left(\genfrac{}{}{0pt}{}{n}{2}\right)$ unique linear classifiers for any random set of $n$ 2-dimensional features (the factor of 2 is due to the fact that for each linear classifier thereare 2 possible assignments of the labelling).

Thus, instead of infinitely many linear classifiers, we realize that as far as a random sample of $n$ training data is concerned, there are at most

$\begin{array}{ccc}\hfill 2\left(\genfrac{}{}{0pt}{}{n}{2}\right)& =& \frac{2n!}{\left(n-2\right)!2!}\hfill \\ & =& n\left(n-1\right)\hfill \end{array}$

unique linear classifiers. That is, using linear classification rules, there are at most $n\left(n-1\right)\approx {n}^{2}$ unique label assignments for $n$ data points. If we like, we can encode each possibility with ${log}_{2}n\left(n-1\right)\approx 2{log}_{2}n$ bits. In $d$ dimensions there are $2\left(\genfrac{}{}{0pt}{}{n}{d}\right)$ hyperplane classification rules which can be encoded in roughly $d{log}_{2}n$ bits. Roughly speaking, the number of bits required for encoding each model is the VC dimension. Theremarkable aspect of the VC dimension is that it is often finite even when $\mathcal{F}$ is infinite (as in this example).

If $\mathcal{X}$ has $d$ dimensions in total, we might consider linear classifiers based on $1,2,\cdots ,d$ features at a time. Lower dimensional hyperplanes are less complex than higher dimensionalones. Suppose we set

$\begin{array}{ccc}\hfill {\mathcal{F}}_{1}& =& \text{linear}\phantom{\rule{4.pt}{0ex}}\text{classifiers}\phantom{\rule{4.pt}{0ex}}\text{using}\phantom{\rule{4.pt}{0ex}}\text{1}\phantom{\rule{4.pt}{0ex}}\text{feature}\hfill \\ \hfill {\mathcal{F}}_{2}& =& \text{linear}\phantom{\rule{4.pt}{0ex}}\text{classifiers}\phantom{\rule{4.pt}{0ex}}\text{using}\phantom{\rule{4.pt}{0ex}}\text{2}\phantom{\rule{4.pt}{0ex}}\text{features}\hfill \\ \hfill \cdots & & \text{and}\phantom{\rule{4.pt}{0ex}}\text{so}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\hfill \end{array}.$

These spaces have increasing VC dimensions, and we can try to balance the empirical risk and a cost function depending on the VC dimension.Such procedures are often referred to as Structural Risk Minimization . This gives you a glimpse of what the VC dimension is all about. In future lectures we will revisit this topic in greaterdetail.

## Hold-out methods

The basic idea of “hold-out” methods is to split the $n$ samples $D\equiv {\left\{{X}_{i},{Y}_{i}\right\}}_{i=1}^{n}$ into a training set, ${D}_{T}$ , and a test set, ${D}_{V}$ .

$\begin{array}{ccc}\hfill {D}_{T}={\left\{{X}_{i},{Y}_{i}\right\}}_{i=1}^{m},& & {D}_{V}={\left\{{X}_{i},{Y}_{i}\right\}}_{i=m+1}^{n}\hfill \end{array}.$

Now, suppose we have a collection of different model spaces $\left\{{\mathcal{F}}_{\lambda }\right\}$ indexed by $\lambda \in \Lambda$ (e.g., ${\mathcal{F}}_{\lambda }$ is the set of polynomials of degree $d$ , with $\lambda =d$ ), or suppose that we have a collection of complexity penalization criteria ${L}_{\lambda }\left(f\right)$ indexed by $\lambda$ ( e.g., let ${L}_{\lambda }\left(f\right)=\stackrel{^}{R}\left(f\right)+\lambda c\left(f\right)$ , with $\lambda \in {\mathbf{R}}^{+}$ ). We can obtain candidate solutions using the training set as follows. Define

$\begin{array}{ccc}\hfill {\stackrel{^}{R}}_{m}\left(f\right)& =& \sum _{i=1}^{m}\ell \left(f\left({X}_{i}\right),{Y}_{i}\right)\hfill \end{array}$

and take

$\begin{array}{ccc}\hfill {\stackrel{^}{f}}_{\lambda }& =& arg\underset{f\in {\mathcal{F}}_{\lambda }}{min}{\stackrel{^}{R}}_{m}\left(f\right)\hfill \end{array}$

or

$\begin{array}{ccc}\hfill {\stackrel{^}{f}}_{\lambda }& =& arg\underset{f\in \mathcal{F}}{min}\phantom{\rule{0.166667em}{0ex}}\left\{{\stackrel{^}{R}}_{m},\left(f\right),+,\lambda ,c,\left(f\right)\right\}\hfill \end{array}.$

This provides us with a set of candidate solutions $\left\{{\stackrel{^}{f}}_{\lambda }\right\}$ . Then we can define the hold-out error estimate using the test set:

$\begin{array}{ccc}\hfill {\stackrel{^}{R}}_{V}\left(f\right)& =& \frac{1}{n-m+1}\sum _{i=m+1}^{n}\ell \left(f\left({X}_{i}\right),{Y}_{i}\right),\hfill \end{array}$

and select the “best” model to be $\stackrel{^}{f}={\stackrel{^}{f}}_{\stackrel{^}{\lambda }}$ where

$\begin{array}{ccc}\hfill \stackrel{^}{\lambda }& =& arg\underset{\lambda }{min}{\stackrel{^}{R}}_{V}\left({\stackrel{^}{f}}_{\lambda }\right)\hfill \end{array}.$

This type of procedure has many nice theoretical guarantees, provided both the training and test set grow with $n$ .

## Leaving-one-out cross-validation

A very popular hold-out method is the so call “leaving-one-out cross-validation” studied in depth by Grace Wahba (UW-Madison,Statistics). For each $\lambda$ we compute

$\begin{array}{ccc}\hfill {\stackrel{^}{f}}_{\lambda }^{\left(k\right)}& =& arg\underset{f\in \mathcal{F}}{min}\frac{1}{n}\sum _{\stackrel{i=1}{i\ne k}}^{n}\ell \left(f\left({X}_{i}\right),{Y}_{i}\right)+\lambda C\left(f\right)\hfill \end{array}$

or

$\begin{array}{ccc}\hfill {\stackrel{^}{f}}_{\lambda }^{\left(k\right)}& =& arg\underset{f\in {\mathcal{F}}_{\lambda }}{min}\frac{1}{n}\sum _{\stackrel{i=1}{i\ne k}}^{n}\ell \left(f\left({X}_{i}\right),{Y}_{i}\right).\hfill \end{array}$

Then we have cross-validation function

$\begin{array}{ccc}\hfill V\left(\lambda \right)& =& \frac{1}{n}\sum _{k=1}^{n}\ell \left({f}_{\lambda }^{\left(k\right)}\left({X}_{k}\right),{Y}_{k}\right)\hfill \\ \hfill {\lambda }^{*}& =& arg\underset{\lambda }{min}V\left(\lambda \right).\hfill \end{array}$

## Summary

To summarize, this lecture gave a brief and incomplete survey of different methods for dealing with the issues of overfitting and modelselection. Given a set of training data, ${D}_{n}={\left\{{X}_{i},{Y}_{i}\right\}}_{i=1}^{n}$ , our overall goal is to find

${f}^{*}=arg\underset{f\in \mathcal{F}}{min}R\left(f\right)$

from some collection of functions, $\mathcal{F}$ . Because we do not know the true distribution ${P}_{XY}$ underlyingthe data points ${D}_{n}$ , it is difficult to get an exact handle on the risk, $R\left(f\right)$ . If we only focus on minimizing the empirical risk $\stackrel{^}{R}\left(f\right)$ we end up overfitting to the training data. Two general approaches were presented.

1. In the first approach we consider an indexed collection of spaces ${\left\{{\mathcal{F}}_{\lambda }\right\}}_{\lambda \in \Lambda }$ such that the complexity of ${\mathcal{F}}_{\lambda }$ increases as $\lambda$ increases, and
$\underset{\lambda \to \infty }{lim}{\mathcal{F}}_{\lambda }=\mathcal{F}.$
A solution is given by
$\begin{array}{c}\hfill {\stackrel{^}{f}}_{{\lambda }^{*}}=arg\underset{f\in {\mathcal{F}}_{{\lambda }^{*}}}{min}{\stackrel{^}{R}}_{n}\left(f\right)\end{array}$
where either ${\lambda }^{*}$ is a function which increases with $n$ ,
$\begin{array}{ccc}\hfill {\lambda }^{*}& =& \lambda \left(n\right),\hfill \end{array}$
or ${\lambda }^{*}$ is chosen by hold-out validation.
2. The alternative approach is to incorporate a penalty term into the risk minimization problem formulation. Here we consideran indexed collection of penalties ${\left\{{C}_{\lambda }\right\}}_{\lambda \in \Lambda }$ satisfying the following properties:
1. ${C}_{\lambda }:\mathcal{F}\to {\mathbf{R}}^{+}$ ;
2. For each $f\in \mathcal{F}$ and ${\lambda }_{1}<{\lambda }_{2}$ we have ${C}_{{\lambda }_{1}}\left(f\right)\le {C}_{{\lambda }_{2}}\left(f\right)$ ;
3. There exists ${\lambda }_{0}\in \Lambda$ such that ${C}_{{\lambda }_{0}}\left(f\right)=0$ for all $f\in \mathcal{F}$ .
In this formulation we find a solution
$\begin{array}{ccc}\hfill {\stackrel{^}{f}}_{{\lambda }^{*}}& =& arg\underset{f\in \mathcal{F}}{min}{\stackrel{^}{R}}_{n}\left(f\right)+{C}_{{\lambda }^{*}}\left(f\right),\hfill \end{array}$
where either ${\lambda }^{*}=\lambda \left(n\right)$ , a function growing the number of data samples $n$ , or ${\lambda }^{*}$ is selected by hold-out validation.

## Consistency

If an estimator or classifier ${\stackrel{^}{f}}_{{\lambda }^{*}}$ satisfies

$E\left[R,\left(,{\stackrel{^}{f}}_{{\lambda }^{*}},\right)\right]\to \underset{f\in \mathcal{F}}{inf}R\left(f\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty ,$

then we say that ${\stackrel{^}{f}}_{{\lambda }^{*}}$ is $\mathcal{F}$ -consistent with respect to the risk $R$ . When the context is clear, we will simply say that $\stackrel{^}{f}$ is consistent.

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