Alternating decision tree

An Alternating Decision Tree (ADTree) is a machine learning methodfor classification. The ADTree data structure and algorithmare a generalization of decision trees and have connections to
boosting. ADTrees were introduced by Yoav Freund and Llew MasonYoav Freund and Llew Mason. TheAlternating Decision Tree Algorithm. Proceedings of the 16thInternational Conference on Machine Learning, pages 124-133 (1999)] .

Motivation

Original boosting algorithms typically combined either decision stumps or decision trees. Boosting decision stumps createsa set of $T$ weighted weak hypotheses (where $T$is the number of boosting iterations), which can be visualized as aset for reasonable values of $T$. Boosting decision treescould result in a final combined classifier with thousands (ormillions) of nodes for modest values of $T$. Both of thesescenarios produce final classifiers in which it is either difficult tovisualize correlations or difficult to visualize at all.Alternating decision trees provide a method for visualizing decisionstumps in an ordered and logical way to demonstrate correlations. Indoing so, they simultaneously generalize decision trees and can beused to essentially grow boosted decision trees in parallel.

Description of the structure

The alternating decision tree structure consists of two components:decision nodes and prediction nodes. Decision nodes specify apredicate condition. Prediction nodes specify a value to add tothe score based on the result of the decision node. Eachdecision node can be seen as a conjunction between aprecondition (the decision node was reached) and the conditionspecified in the decision node.

Perhaps the easiest way to understand the interaction of decision andprediction nodes is through an example.The following example is taken from JBoost performing boosting for 6 iterations on the [http://www.ics.uci.edu/~mlearn/databases/spambase/ spambase dataset] (available from the [http://www.ics.uci.edu/~mlearn/MLRepository.html UCI Machine Learning Repository] ).Positive examples indicate that the message is spam and negativeexamples are not spam. During each iteration, a single node is added to the ADTree. The ADTree determined by the learning algorithm implemented in JBoost is:

center|800px|An ADTree for 6 iterations on theSpambase dataset.

The tree construction algorithm is described below in theDescription of the algorithm section. We now show how tointerpret the tree once it has been constructed. We focuson one specific instance:

For this instance, we obtain a score that determines theclassification of the instance. This score not only acts as aclassification, but also as a measure of confidence. The actual orderthat the ADTree nodes are evaluated will likely be different then theorder in which they were created. That is, the node from iteration 4can be evaluated before the node from iteration 1. There areconstraints to this (e.g. node from iteration 2 must be evaluatedbefore the node from iteration 5). In general, either breadth-firstor depth-first evaluation will yield the correct interpretation.

The following table shows how the score is created (progressivescore) for our above example instance:

There are a few observations that we should make
* The final classification of the example is positive (0.657), meaning that the example is considered to be spam.
* All nodes at depth 1 have their predicate evaluated and one of their prediction nodes contributes to the score. Thus a tree with depth 1 is the equivalent of boosted decision stumps.
* If a decision node is not reached (the node from iteration 5 in the above example) then the node's predicate and subsequent prediction nodes will not be evaluated.

Description of the algorithm

The alternating decision tree learning algorithm is described in theoriginal paper. The general idea involves afew main concepts:
* The root decision node is always TRUE or FALSE
* The tree is grown iteratively. The total number of iterations is generally decided prior to starting the algorithm.
* Each decision node ($c\_2$) is selected by the algorithm based on how well it discriminates between positive and negative examples.
* Once a decision node is created, the prediction node is determined by how well the decision node discriminates.

Before the algorithm, we first define some notation. Let $c$ be a predicate, then
* $W\_+(c)$ is the weight of all positively labeled examples that satisfy $c$
* $W\_-(c)$ is the weight of all negatively labeled examples that satisfy $c$
* $W(c)$ is the weight of all examples that satisfy $c$
* We call $c$ a precondition when it is a conjunction of previous base conditions and negations of previous base conditions

The exact algorithm is:

INPUT: $m$ examples and labels$(x\_1,y\_1),ldots,(x\_m,y\_m)$

Set the weight of all examples to $W\_1(x\_j)\; =\; 1/m$

Set the margin of all examples to $r\_1(x\_j)\; =\; 0$

The root decision node is always $c=$TRUE, with a single prediction node

$a=frac\{1\}\{2\}\; extrm\{ln\}frac\{W\_+(T)\}\{W\_-(T)\}$

For $t\; =\; 1,\; ldots,\; T$ do:
* Let $c\_1\; in\; P\_t$ be a precondition (that is, the node being created can be reached via $c\_1$) and $c\_2$ be a condition (the new node). Then each decision node ($c\_2$) is selected by the algorithm based on how well it discriminates between positive and negative examples. The original ADTree algorithm minimizes the criterion $2\; left(\; sqrt\{W\_+(c\_1wedge\; c\_2)\; W\_-(c\_1\; wedge\; c\_2)\}\; +\; sqrt\{W\_+(c\_1wedge\; eg\; c\_2)\; W\_-(c\_1\; wedge\; eg\; c\_2)\}\; ight)\; +W(\; eg\; c\_2)$.
* Once a decision node is created, the prediction nodes are determined by $a=frac\{1\}\{2\}\; extrm\{ln\}frac\{W\_+(c\_1wedge\; c\_2)\}\{W\_-(c\_1\; wedge\; c\_2)\}$ and $b=frac\{1\}\{2\}\; extrm\{ln\}frac\{W\_+(c\_1wedge\; eg\; c\_2)\}\{W\_-(c\_1\; wedge\; eg\; c\_2)\}$
* Add the conditions $c\_1\; wedge\; c\_2$ and $c\_1\; wedge\; eg\; c\_2$ to the set of possible preconditions $P\_\{t+1\}$
* Update the weights: $W\_\{t+1\}(x\_j)\; =\; W\_\{t\}(x\_j)e^\{r\_t(x\_j)y\_j\}$

Empirical Results

Figure 6 in the original paper demonstrates thatADTrees are typically as robust as boosted decision trees and boosted decision stumps.

References

External links

* [http://www.cs.ucsd.edu/~aarvey/jboost/presentations/BoostingLightIntro.pdf An introduction to Boosting and ADTrees] (Has many graphical examples of alternating decision trees in practice)