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\begin{document}

\renewcommand{\baselinestretch}{0.98} 
\twocolumn[
\icmltitle{Combining Train Set and Test Set Bounds}

\icmlauthor{John Langford}{jcl@cs.cmu.edu}
\icmladdress{Computer Science Department, Carnegie-Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15217}
\vskip 0.3in
]

\begin{abstract}
This paper is about bounds on future error rates. We present a theorem for combining
an arbitrary test set based bound with an arbitrary training set based bound.
Appropriate use of this theorem results in a combined bound with two properties:
1) the combined bound is never much worse than either the training set based
bound or the test set based bound and 2) the combined bound is sometimes better
than either bound individually. Empirical validation is presented showing the
effectiveness of the combined bound.
\end{abstract}

\section{Introduction}

Traditionally, there are two broad classes of techniques for bounding the future
error rate of a learned classifier under an assumption that all examples are
drawn iid from an unknown distribution. These techniques are ``training set''
based techniques and ``test set'' based techniques. Each approach has both
disadvantages and advantages relative to the other. The message of this paper
is that the techniques are \emph{not} exclusive. The two approaches can be freely
blended to construct a bound which is ``better'' (in some sense) than either
bound individually. 

Test set based techniques include bagging, cross-validation, and the simple
holdout set. The primary advantage of test set based techniques is that they
work: typically they can be used to give a tight upper bound on the future error
rate of some hypothesis. There are, however, several drawbacks to the test set
approach. The largest drawback is that data used for testing can not be used
for training. This can be a very serious issue when learning problems exhibit
``phase transitions'' where a few extra examples suddenly make the chosen
classifier much more accurate. If these extra examples are in the holdout set,
our learning algorithm will produce a poor classifier. Another, drawback of
test set based techniques is that they are not always well-analyzed. Of the
above approaches, only the behavior of the holdout set is well understood on
arbitrary learning algorithms. 

Training set based techniques include the famous VC analysis \cite{VC}, and
is the focus of much of the work in computational learning theory. The biggest
advantage of training set based techniques is that all examples can be used
for both learning and bound construction. The drawback of training set based
techniques is that they are often too loose to provide any useful information.
Sometimes training set based techniques are actually tighter than test set based
techniques. However this has not been used in practice, principally because
it is difficult to predict in advance whether or not a training set based technique
will be tight enough to yield interesting information. 

The combination theorem proved in this paper makes \emph{all} computable training
set based true error bound techniques practically useful. The prescription for
use is straightforward: 

\begin{enumerate}
\item Choose a training set bound, a holdout set, and a combined train set/test set
bound.
\item Train on the training set to choose a classifier.
\item Evaluate errors on the train and the test sets and use the combined train set/test
set bound to report a high-probability bound on the future error rate. 
\end{enumerate}
A judicious choice of combined bound can be used with the simple guarantee that
the combined bound is never much worse than the lowest of the training set and
test set bounds. In addition, we can also guarantee that the combined bound
is sometimes better than either individual bound. 

The remainder of this paper first develops the combined bound theorem, and then
presents empirical results showing how this technique can be useful. 


\section{How to Combine Training and Testing Bounds}


\subsection{Setup and Motivation}

We will define a classification problem as a distribution, \( D \), on a space
\( X\times \{0,1\} \). Given a set \( S \) of \( |S|=m \) draws of labeled
examples \( (x,y) \) from the distribution \( D \), the goal of any learning
algorithm is to choose a hypothesis, \( h:X\rightarrow \{0,1\} \), with a low
true error rate, \( e_{D}(h)=\Pr _{x,y\sim D}(h(x)\neq y) \). Unfortunately,
the distribution \( D \) is unknown and so the true error rate, \( e_{D}(h) \)
is not evaluatable. Nonetheless, it is often possible to bound the true error
rate in terms of observable quantities such as the empirical (or training) error
\( \hat{e}_{S}(h)=\Pr _{x,y\sim U(S)}(h(x)\neq y) \). Here \( U(S) \) is the
uniform distribution on the set \( S \). It will be important to make a distinction
between train and test sets. Test sets, \( S_{\textrm{test}} \) will always
have a subscript ``test'' and the number of test set examples will be denoted
by \( m_{\textrm{test}} \). For brevity, we will denote the empirical error
on a test set as \( \hat{e}_{\textrm{test}}(h) \).

Given a fixed classifier and a classification problem, the distribution of a
test error on \( m \) labeled examples is simple. The probability of an error
for each example is \( e_{D}(h) \) and independence implies the empirical error
will be Binomially distributed. In particular, the Binomial distribution is
given by: 
\[
\begin{array}{ccc}
\textrm{Bin}(m,k,p) & \equiv  & \sum _{j=0}^{k}\binom{m}{j}p^{j}(1-p)^{m-j}\\
 & 
\end{array}\]
and note that: 


\[
\textrm{Bin}(m,k,p)=\Pr _{S_{\textrm{test}}\sim D^{m_{\textrm{test}}}}\left( \hat{e}_{\textrm{test}}(h)\leq \frac{k}{m_{\textrm{test}}}|e_{D}(h)\right) \]


Given a fixed learning algorithm and learning problem, the training error will
have a considerably more complicated distribution. We can nonetheless, regard
the training error as a fixed random variable which has some cumulative distribution
parameterized by many parameters, one of which is the true error rate of the
output hypothesis (which is itself a random variable). 

How can we construct a confidence interval based upon information from both
the training and testing sets? There are several possibilities.

\begin{enumerate}
\item Construct an interval based upon the probability that \emph{both} true error
upper bounds are violated. 
\item Construct an interval based upon the probability that at least one of the true
error upper bounds is violated.
\item Use a more general technique.
\end{enumerate}
Figure \ref{fig-and} represents Technique (1) visually. 

\begin{figure}
{\par\centering \resizebox*{0.9\columnwidth}{!}{\includegraphics{thesis/conf_1.eps}} \par}


\caption{\label{fig-and} This is a graph representing the rejection region for a combined
bound based upon both bounds being violated. The marked region is the low probability
events disallowed.}
\end{figure}


The essential problem with technique (1) is that the resulting true error bound
is the \emph{maximum} (minus a small amount) of the bounds based upon both the
test set and the training set. Given that we don't trust the training set based
bound to always be tight, we expect this combination not behave well. 

Technique (2) can be seen visually in Figure \ref{fig-or}.

\begin{figure}
{\par\centering \resizebox*{0.9\columnwidth}{!}{\includegraphics{thesis/conf_2.eps}} \par}


\caption{\label{fig-or} This is a graph representing the rejection region for a combined
bound based upon either bound being violated. Note that the ``width'' of the
one violation area is smaller than in figure \ref{fig-and}. This occurs because
the probability mass of the rejection region stays constant.}
\end{figure}


Technique (2) works moderately well. Mathematically, we can calculate
the minimum of the two error bounds and add a small amount. This
approach is nearly equivalent to taking a union bound. While this
approach allows us to combine the bounds, it does not let us achieve
an improvement over either bound individually. This property is
intuitively possible. Certainly, if we use two test sets, the
confidence interval should improve.

A better approach is possible. We would like to construct a combined bound with
the form given in Figure \ref{fig-other}.

\begin{figure}
{\par\centering \resizebox*{0.9\columnwidth}{!}{\includegraphics{thesis/conf_4.eps}} \par}


\caption{\label{fig-other} This is a graph representing a possible rejection region
for some other combined bound.}
\end{figure}
 Such a rejection region has two important properties:

\begin{enumerate}
\item If one bound is loose, it does not greatly harm the final true error bound. 
\item The final true error bound can be tighter than either individual true error
bound. 
\end{enumerate}
Showing that technique (2) works is just an application of the union bound.
Given any two bounds on the true error rate, we can apportion \( \frac{\delta }{2} \)
confidence to each bound. Then both bounds will hold with probability \( \delta  \)
which implies that the minimum of the two true error bounds (worsened by the
substitution \( \delta \rightarrow \frac{\delta }{2} \)) holds.

One interesting possibility to consider is the rejection region of two test
set based bound. The \emph{standard} rejection region for the combination of
two test set bounds will be a diagonal orthogonal to the identity as in Figure
\ref{fig-test-shape}. 
\begin{figure}
{\par\centering \resizebox*{0.9\columnwidth}{!}{\includegraphics{thesis/conf_3.eps}} \par}


\caption{\label{fig-test-shape} This is a graph showing the bound constructed by combining
two (asymmetrically sized) test sets. Since the total number of errors is a
sufficient statistic for bound calculations, the boundary of the bound violation
region is at a constant number of errors which is perpendicular to the identity.}
\end{figure}
 A bound based upon a rejection region of this form is desirable because it
can result in significant improvements. The combined bound we develop mixes
the possible improvement of Figure \ref{fig-test-shape} with the soft minimum
of \ref{fig-or}. The soft minimum is necessary in order to protect against
the (sometimes large) pessimism of training set based bounds. 


\subsection{General Approaches for Combined Bounds}

Showing that a more general technique works must start with a discussion of
confidence intervals. Fundamentally, a bound can be viewed as a set of outcomes.
Let \( X \) be a space of outcomes, then a bound \( \phi \subseteq X \) is
a subset. The probability that this generalized bound is violated for some distribution
\( P \) is given by: 
\[
\Pr _{x\sim P}(x\in \phi )\]
Note that the rejection region \( \phi  \) can be parameterized by both the
distribution \( P \) and confidence \( \delta  \) to get: 
\[
\Pr _{x\sim P}(x\in \phi _{P}(\delta ))\leq \delta \]
We can expand the definition of a bound to include a \emph{randomized} rejection
region. In particular, let \( \phi _{P}(w,\delta ) \) satisfy: 
\[
\forall w:\, \, \Pr _{x\sim P}(x\in \phi _{P}(w,\delta ))\leq \delta \]
 Then, the following statement holds:
\[
\Pr _{w\sim Q,x\sim P}(x\in \phi _{P}(w,\delta ))\leq \delta \]
In fact, we can make a stronger statement. If 
\begin{equation}
\label{eq-randomized}
E_{w\sim Q}\Pr _{x\sim P}(x\in \phi _{P}(w,\delta ))\leq \delta 
\end{equation}
 then 
\begin{equation}
\label{eq-pr_randomized}
\Pr _{w\sim Q,x\sim P}(x\in \phi _{P}(w,\delta ))\leq \delta 
\end{equation}
. 

Randomized confidence intervals are useful here because we can regard the draw
of the ``test'' set as constructing a randomized interval for the ``train''
set. As long as constraint \ref{eq-randomized} is obeyed, the bound will hold
with probability at least \( \delta  \). We can then take a supremum over all
classification problems \( D \) and the bound will then yield the following
theorem:

\begin{thm}
\label{th-tnt} (Exact test and train bound) Let \( \phi _{D}(\delta ) \) be
any bound satisfying: 
\[
\forall D\, \, \forall \delta \geq 0\, \, \Pr _{S\sim D^{m}}(S\in \phi _{D}(\delta ))\leq \delta \]
Let \( f(S_{\textrm{test}}) \) be any function satisfying: 
\[
E_{S_{\textrm{test}}\sim D^{m_{\textrm{test}}}}\Pr _{S\sim D^{m}}(S\in \phi _{D}(f(S_{\textrm{test}})*\delta ))\leq \delta \]
 then:
\[
\Pr _{S,S_{\textrm{test}}\sim D^{m+m_{\textrm{test}}}}(S\in \phi _{D}(f(S_{\textrm{test}})*\delta ))\leq \delta \]

\end{thm}
\begin{proof}
This is a simple application of the definition of probability.
\end{proof}
At this point we are (in some sense) ``done''---we can flexibly combine any
train set based bound with the holdout bound and verify that it holds using
an exact calculation of the expectation, 
\[
E_{S_{\textrm{test}}\sim D^{m_{\textrm{test}}}}\Pr _{S\sim D^{m}}(S\in \phi _{D}(f(S_{\textrm{test}})*\delta ))\leq \delta \]


There are many possible choices of the function \( f(S_{\textrm{test}}) \),
each of which leads to a different combined training and testing bound. In particular,
there exists an \( f \) for the conjunction and disjunction methods---cases
(1) and (2) in the previous section. We can also design an \( f \) which satisfies
our goals of ``never much worse and sometimes better''.

Nonetheless, some more effort is required because in practice it is sometimes
desirable to use a bound for the holdout set rather than an exact calculation
of the expectation. 


\subsection{Approximations in Combinations}

The inexact nature of bounds forces us to impose a monotonic structure on the
function \( f(S_{\textrm{test}}) \). For simplicity, we will restrict to functions
of the form \( f(\hat{e}_{\textrm{test}}(h)) \) where \( \hat{e}_{\textrm{test}}(h) \)
is the test error on hypothesis \( h \). This simplification is not necessary
and this technique can be extended to arbitrary test set based techniques.

We can consider any upper bound \( \theta _{D}(\delta ) \) on the true error
rate, \( e_{D}(h) \), as inducing a cumulative distribution on the test set
events. 

This cumulative distribution is \emph{not} the cumulative distribution of the
underlying (Binomial) probability. To construct this distribution, let:
\[
F_{\theta }(\hat{e}_{\textrm{test}}(h))=\inf \{\delta :\hat{e}_{\textrm{test}}(h)\in \theta _{D}(\delta )\}\]
Intuitively, \( F_{\theta }(\hat{e}_{\textrm{test}}(h)) \) is the smallest
\( \delta  \) such that the test error \( \hat{e}_{\textrm{test}}(h) \) is
rejected. 

\begin{lem}
The function \( F_{\theta }(\hat{e}_{\textrm{test}}(h)) \) is a cumulative
distribution function.
\end{lem}
\begin{proof}
In order to show that the function is a cumulative distribution function, we
must show that it varies between \( 0 \) and \( 1 \) for all values of \( \hat{e}_{\textrm{test}}(h) \).
Since \( \theta _{D}(\delta ) \) is an upper bound, the following inequality
holds: 
\[
\forall S_{\textrm{test}}\, \, F_{\theta }(\hat{e}_{\textrm{test}}(h))\geq \textrm{Bin}(m,m*\hat{e}_{\textrm{test}}(h),e_{D}(h))\]
 This inequality implies the value of \( F_{\theta }(\hat{e}_{\textrm{test}}(h)) \)
is always at least as large as the CDF of the underlying Binomial distribution.
Note, that different \( S_{\textrm{test}} \) are implicitly aliased under this
technique. We also have the inequality \( F_{\theta }(\hat{e}_{\textrm{test}}(h))\leq 1 \)
because all true error rate upper bounds are vacuous above a true error rate
bound of \( 1 \).
\end{proof}
We have shown that \( F_{\theta } \) is a cumulative distribution function
over the value of the empirical error. Given the upper bound cumulative, \( F_{\theta } \),
we can look at distributions satisfying: 
\[
E_{\hat{e}_{\textrm{test}}(h)\sim F_{\theta }}\Pr _{S\sim D^{m}}(S\in \phi (f(\hat{e}_{\textrm{test}}(h))*\delta ))\leq \delta \]
 If we are guaranteed that \( f(\hat{e}_{\textrm{test}}(h)) \) decreases monotonically
then equation \ref{eq-pr_randomized} will hold. This is the essence of our
theorem.

\begin{thm}
\label{th-mtnt} (Approximate test and train bound) Let \( \phi _{D}(\delta ) \)
be any bound satisfying 
\[
\forall D\, \, \forall \delta >0\, \, \Pr _{S\sim D^{m}}(S\in \phi _{D}(\delta ))\leq \delta \]
. Let \( f(\hat{e}_{\textrm{test}}(h)) \) be any monotonic decreasing function
satisfying: 
\[
E_{\hat{e}_{\textrm{test}}(h)\sim F_{\theta }}\Pr _{S\sim D^{m}}(S\in \phi _{D}(f(\hat{e}_{\textrm{test}}(h))*\delta ))\leq \delta \]
 , then:
\[
\Pr _{S,S_{\textrm{test}}\sim D^{m+m_{\textrm{test}}}}(S\in \phi _{D}(f(\hat{e}_{\textrm{test}}(h))*\delta ))\leq \delta \]

\end{thm}
\begin{proof}
Note that \( \Pr _{S\sim D^{m}}(S\in \phi _{D}(f(\hat{e}_{\textrm{test}}(h))*\delta )) \)
is a monotonic decreasing function of \( \hat{e}_{\textrm{test}}(h) \). For
any monotonic decreasing function \( g(x) \) and any two cumulative distribution
functions \( F_{1}(x) \) and \( F_{2}(x) \) satisfying \( \forall x\, \, F_{1}(x)\leq F_{2}(x) \)
we have:
\[
E_{x\sim F_{1}(x)}g(x)\leq E_{x\sim F_{2}(x)}g(x)\]
Let \( F(x) \) be the cumulative distribution of the Binomial and note that
the definition of a bound implies: \( \forall x\, \, F_{\theta }(x)\geq F(x) \).
Applying these inequalities, we get: 
\[
\delta \geq E_{\hat{e}_{\textrm{test}}\sim F_{\theta }}\Pr _{S\sim D^{m}}(S\in \phi _{D}(f(\hat{e}_{\textrm{test}}(h))*\delta ))\]
\[
\geq E_{\hat{e}_{\textrm{test}}\sim F}\Pr _{S\sim D^{m}}(S\in \phi _{D}(f(\hat{e}_{\textrm{test}}(h))*\delta ))\]
 Given this, an application of theorem \ref{th-tnt} completes the proof.
\end{proof}
The only constraint that we must check in applying a combined train and test
bound is the monotonicity constraint. Heuristically, this is satisfied for the
functions graphed in figures \ref{fig-and}, \ref{fig-or}, \ref{fig-other},
and \ref{fig-test-shape} since the set of excluded events increases monotonically
along the \( x \) axis as it decreases along the \( y \) axis. 

It is worth noticing that this approach applies to \emph{any} training set based
bound which holds for all \( \delta >0 \) and all learning problems \( D \).
In particular, it holds for VC-dimension based bound \cite{VC} although we
do not use that approach here.


\subsection{The train and test bound}

An explicit mathematical form for a combined train and test bound can be given
by considering the bound-based cumulative distributions, \( F_{\theta } \),
and a similar distribution for the training set, \( F_{\phi } \). In particular,
we can define the rejection region to be 
\[
\{S_{\textrm{test}},S:F_{\theta }(S_{test})F_{\phi }(S)\leq t(\delta )\}\]
 where \( t(\delta ) \) is a function satisfying 
\[
\Pr _{S_{\textrm{test}},S\sim F_{\theta },F_{\phi }}(F_{\theta }(\hat{e}_{\textrm{test}}(h))F_{\phi }(S))\leq t(\delta))\leq \delta \]
. The monotonic constraint is satisfied by this construction because \( F_{\theta } \)
is implicitly monotonic decreasing with \( F_{\phi }(S) \) given a constant
\( t \). We will use this bound for combining train and test sets in the experiments. 

Calculation of this bound is straightforward. Essentially, we do a binary search
for the true error rate \( e(h) \) which places our observed test error and
training error on the boundary of the rejection region. Since \( e(h) \) is
in the interval \( [0,1] \), each tested true error rate will increase the
precision of the true error bound calculation by 1 bit. The computation can
be halted at the machine precision or some other negligible size.

By observation, this combined bound will satisfy our desirable properties (1)
``never hurt much'' and (2) ``sometimes help''. 


\section{Experimental Results: Decision Trees}


\subsection{The Approach and Bounds used}

We will test and compare various bounds on an ID3 based decision tree algorithm
using discrete datasets from the UCI database of machine learning problems.
The exact details of the decision tree and bound implementations are not discussed
here although the details are quite important in order to replicate these results.
For full details see \cite{thesis} and note that a true error bound calculation
program is available \cite{program}.

For every dataset, our goal is to learn a hypothesis \( h \) and a high-probability
upper bound on the future error rate of that hypothesis. We wish to find a hypothesis
with the smallest possible high-probability upper bound. For our purposes, there
are 3 varieties of bounds: Training set bounds, test set bounds, and combined
training and testing set based bounds. We will compare these bounds in 3 ways:

\begin{enumerate}
\item Test set bounds (figure \ref{fig-test}): Train on \( 80\% \) of the data then
use the remaining \( 20\% \) for the test set bound.  
\item Training set bounds (figure \ref{fig-upper-error}): Train on \emph{all} the
data and then calculate the training set based bound. 
\item Train and Test bounds (figure \ref{fig-holdout-train}): Train on \( 80\% \)
of the data then calculate a bound using performance on both the training and
the testing sets.
\end{enumerate}
We choose this comparison because it is realistic. Anyone attempting to apply
machine learning techniques faces a choice of train set/test set example allocation
and the choices made here are not uncommon. 

The training set based bounds are the following:

\begin{enumerate}
\item The discrete hypothesis bound such as appears in \cite{Tom} formed by simply
counting the number of decision trees.
\item The Microchoice bound (first introduced in \cite{MC} but see \cite{thesis}
for improvements). The Microchoice bound can (almost) be thought of as a particular
choice of description language for the Occam's Razor bound \cite{occam}.
\item The Shell bound \cite{Shell} (\cite{thesis} for improvements) and the Sampled
Shell bound \cite{thesis}. The shell bound is a functional improvement on the
Occam's Razor style bounds which requires significant computation in order to
evaluate.
\end{enumerate}
We only use the holdout bound (see \cite{thesis} chapter 2 for details) for
test sets because other test set based bounds lack a solid analysis on decision
trees. We also compare two combined bounds:

\begin{enumerate}
\item A combined training and testing bound using the holdout bound and microchoice
bound.
\item A combined training and testing bound using the holdout bound and the (stochastic)
shell bound.
\end{enumerate}
The datasets (see figure \ref{fig-counts} for sizes) consist of 13 UCI database
discrete datasets which appeared easiest to use with a decision tree. It is
important to note that the problems were not chosen because they optimized particular
bounds well, nor were any of the bounds ``tuned'' to do better on any of these
datasets. 

For all bounds, we use a probability of failure of \( \delta =0.1 \). The exact
data generating Figure \ref{fig-upper-error} and Figure \ref{fig-holdout-train}
are listed in the appendix.

\begin{figure}
{\par\centering \resizebox*{0.9\columnwidth}{!}{\rotatebox{270}{\includegraphics{thesis/count.ps}}} \par}


\caption{\label{fig-counts} A graph showing the number of examples in each problem.
Note that holdout sets use only \protect\( 20\%\protect \) of the available
data. }
\end{figure}


\vspace{0.3cm}
{
\begin{figure}
{\par\centering \resizebox*{0.9\columnwidth}{!}{\rotatebox{270}{\includegraphics{thesis/test.ps}}} \par}


\caption{\label{fig-test} This is a graph of the confidence intervals implied by the
holdout bound (See the introduction of \cite{thesis} for details). In this
figure (and all others) we use \protect\( \delta =0.1\protect \) bounds for
each tail so the confidence intervals hold with probability \protect\( 0.8\protect \)
over the data set. Note that an estimate of the size of a the holdout set can
be inferred by the width of the confidence interval.}
\end{figure}
\begin{figure}
{\par\centering \resizebox*{0.95\columnwidth}{!}{\rotatebox{270}{\includegraphics{thesis/train_upper.ps}}} \par}


\caption{\label{fig-upper-error} This graph compares upper bounds based upon the training
set. The left column is a discrete hypothesis bound, the middle column is a
microchoice bound, and the right column is a shell bound. The existence of a
dashed line implies an approximate calculation of the stochastic shell bound.}
\end{figure}
\begin{figure}
{\par\centering \resizebox*{0.95\columnwidth}{!}{\rotatebox{270}{\includegraphics{thesis/test_upper.ps}}} \par}


\caption{\label{fig-holdout-train} This graph compares the true error upper bound for
test set (left column) and two combined bounds based upon Microchoice+holdout
(middle column) and Shell+holdout (right column) bounds. Once again, a dashed
line implies an approximate calculation.}
\end{figure}
\par}
\vspace{0.3cm}


\subsection{Discussion}

It is difficult to answer the question ``which bound is tighter?'' in a theoretical
way. In fact, all of the training set based bounds we use \emph{could} be the
``best'' depending on the exact learning problem (and algorithm). For example,
the Microchoice bound is worse than the Simple bound when the hypothesis chosen
happens to be one of the ``last'' hypotheses with a long description length.
The results in Figure \ref{fig-upper-error} show there is no total ordering
amongst the bounds although there is a noticeable rough ordering:  


\[
\textrm{Simple}>\textrm{Microchoice }>\textrm{Shell}\simeq \textrm{Holdout}\]
 This ordering is approximately as expected based on theoretical considerations.
 The Simple bound can never be much better than Microchoice bound and the Microchoice
bound can be arbitrarily tighter than the Simple bound.  A similar statement
holds for the Microchoice Bound and the Shell bound.   The Shell bound is not
always the best, but it does behave well in comparison to the more standard
holdout approach. 

Empirically, we can observe a very noticeable behavior.  For problems with less
than \( 100 \) examples the training set based bounds are superior to the holdout
bound.  Between \( 100 \) and \( 1000 \) examples, the behavior changes with
the holdout bound generally winning, although not necessarily by much.  Above
\( 1000 \) examples, the holdout bound is significantly and consistently tighter
than the training set based bounds.  This behavior is expected because the training
set based bounds are typically loose.  In particular, the problem of correlated
hypotheses has yet to be solved in a convincing manner on discrete hypothesis
spaces. Once the holdout set becomes large enough to achieve statistical certainty,
the training set based bounds can not compete.

The combined bounds are compared with the holdout bound in \ref{fig-holdout-train}
and the results are the real import of this paper. By comparison, we have the
following rough ordering:
\[
\textrm{Holdout}>\textrm{Holdout}+\textrm{Micro}>\textrm{Holdout}+\textrm{Shell}\]


These improvements are typically most significant when the bound for
the test set is weak due to few test examples, but improvement can and
does occur even when this is not the case.  For example, this occurs
with the ``shroom'' problem.

The combined approach appears to have the best behavior in practice on both
large and small datasets. By examination, we can also see that the two guarantees
stated in the abstract hold as well: 

\begin{enumerate}
\item The combined bound is never much worse than the best bound.
\item The combined bound is is sometimes (a little) better than either bound.
\end{enumerate}

\section{Conclusion}

We proved a theorem allowing us to combine train and test set based true error
bounds in a general manner. Then, we picked a particular technique for combining
the train and test bounds and applied them to a decision trees learned on discrete
UCI database problems. The results show that the combined approach is generally
better on these experiments. 

One significant implication of the combined bound approach is that it makes
theoretical improvements in training set based bounds much more relevant for
practical use. This is important because it connects significant theoretical
work with practical use.

There are several directions of future investigation which could further strengthen
any of these approaches. For the training set based bounds, finding a quantitatively
useful bound which takes into account correlation between hypotheses remains
an open problem. Note that VC-bound and associated covering number approaches
\cite{VC} address this, but not in a manner that results in a bound which is
satisfying for practical use.

We tested the simplest of holdout techniques so another natural extension is
to test other holdout techniques.  This was not done here, because the theory
of these other techniques is lacking and contributions there could be of great
import.

The particular method we used to combine training and testing set bounds was
a simple choice. It is easy to imagine that other combinations (other choices
of the function \( f \)) can be beneficial. 

\begin{thebibliography}{1}
\bibitem{occam}A. Blumer, A. Ehrenfeucht, D. Haussler, M. Warmuth. ``Occam's Razor.'' Information
Processing Letters 24: 377-380, 1987.
\bibitem{thesis}John Langford, Thesis ``Quantitatively Tight Sample Complexity Bounds'', Forthcoming,
draft available at: http://www.cs.cmu.edu/\~{}jcl/research/thesis/thesis.ps
\bibitem{program}John Langford, bound calculation programs, http://www.cs.cmu.edu/\~{}jcl/programs/bound/bound.html
\bibitem{MC}John Langford and Avrim Blum 1999. Microchoice Bounds and Self Bounding learning
algorithms. COLT99. http://www.cs.cmu.edu/\~{}jcl/papers/microchoice/mc.ps
\bibitem{Shell}John Langford and David McAllester, ``Computable Shell Decomposition Bounds''
COLT 2000.
\bibitem{Tom}Tom Mitchell, ``Machine Learning'' McGraw Hill 1997.
\bibitem{VC}V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative
frequencies of events to their probabilities. Theory of Probab. and its Applications,
16(2):264-280, 1971. 
\end{thebibliography}

\section{Appendix: Exact results}

Here we present the exact results behind the graphs. This first table lists
the upper bounds associated training set bounds and the holdout set bound.

\vspace{0.3cm}
{\centering \begin{tabular}{|c|c|c|c|c|c|}
\hline 
Problem&
Number&
Simple&
Micro&
Shell&
Holdout\\
\hline 
\hline 
adult&
20&
\( 0.579 \)&
\( 0.605 \)&
\( 0.447 \)&
\( 0.438 \)\\
\hline 
shroom&
8124&
\( 1.0 \)&
\( 0.083 \)&
\( 0.016 \)&
\( 0.0014 \)\\
\hline 
audio&
200&
\( 1.0 \)&
\( 0.922 \)&
\( 0.865 \)&
\( 0.838 \)\\
\hline 
balance&
625&
\( 0.933 \)&
\( 0.405 \)&
\( 0.409 \)&
\( 0.287 \)\\
\hline 
car&
1728&
\( 0.926 \)&
\( 0.303 \)&
\( 0.269 \)&
\( 0.141 \)\\
\hline 
votes&
435&
\( 1.0 \)&
\( 0.180 \)&
\( 0.097 \)&
\( 0.06 \)\\
\hline 
krkp&
3196&
\( 1.0 \)&
\( 0.171 \)&
\( 0.079 \)&
\( 0.026 \)\\
\hline 
lung&
32&
\( 1.0 \)&
\( 0.963 \)&
\( 0.051 \)&
\( 0.907 \)\\
\hline 
nursery&
12960&
\( 0.91 \)&
\( 0.181 \)&
\( 0.093 \)&
\( 0.013 \)\\
\hline 
postop&
90&
\( 1.0 \)&
\( 0.498 \)&
\( 0.475 \)&
\( 0.671 \)\\
\hline 
shuttle&
15&
\( 1.0 \)&
\( 0.908 \)&
\( 0.805 \)&
\( 0.965 \)\\
\hline 
soybean&
307&
\( 1.0 \)&
\( 0.945 \)&
\( 0.879 \)&
\( 0.811 \)\\
\hline 
yellow&
16&
\( 1.0 \)&
\( 0.902 \)&
\( 0.822 \)&
\( 0.965 \)\\
\hline 
\end{tabular}\par}
\vspace{0.3cm}

The next table presents the results of holdout and combined bounds.

\vspace{0.3cm}
{\centering \begin{tabular}{|c|c|c|c|}
\hline 
Problem&
Holdout&
Holdout+Micro&
Holdout+Shell\\
\hline 
\hline 
adult&
\( 0.438 \)&
\( 0.438 \)&
\( 0.438 \)\\
\hline 
shroom&
\( 0.0014 \)&
\( 0.0014 \)&
\( 0.0014 \)\\
\hline 
audio&
\( 0.838 \)&
\( 0.864 \)&
\( 0.862 \)\\
\hline 
balance&
\( 0.287 \)&
\( 0.313 \)&
\( 0.313 \)\\
\hline 
car&
\( 0.141 \)&
\( 0.155 \)&
\( 0.129 \)\\
\hline 
votes&
\( 0.06 \)&
\( 0.072 \)&
\( 0.074 \)\\
\hline 
krkp&
\( 0.026 \)&
\( 0.026 \)&
\( 0.030 \)\\
\hline 
lung&
\( 0.907 \)&
\( 0.871 \)&
\( 0.853 \)\\
\hline 
nursery&
\( 0.013 \)&
\( 0.013 \)&
\( 0.0100 \)\\
\hline 
postop&
\( 0.671 \)&
\( 0.446 \)&
\( 0.450 \)\\
\hline 
shuttle&
\( 0.965 \)&
\( 0.791 \)&
\( 0.778 \)\\
\hline 
soybean&
\( 0.811 \)&
\( 0.833 \)&
\( 0.834 \)\\
\hline 
yellow&
\( 0.965 \)&
\( 0.789 \)&
\( 0.740 \)\\
\hline 
\end{tabular}\par}
\vspace{0.3cm}

\end{document}
