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A General Framework for Fast and Accurate Regression by Data
Summarization in Random Decision Trees
Wei Fan, IBM T.J.Watson
Joe McCloskey, US Department of Defense
Philip Yu, IBM T.J.Watson
Three DM Problems
Classification: Label: given set of labels in training data.
Probability Estimation: Similar to the above setting: estimate the
probability that x is an example of class y. Difference: no truth is given, i.e., no true
probability Regression:
Target value: continuous values.
Model Approximation True model or correct model.
Generates y for each x with probability P(y|x). Normally never known in reality.
Perfect model: never makes mistakes or has the same prediction as the true model.
Not always possible due to: Stochastic nature of the problem Noise in training data Data is insufficient
Optimal Model Loss function L(t,y) to evaluate performance.
Optimal decision decision y* is the label that minimizes expected loss when x is sampled repeatedly:
Examples 0-1 loss: y* is the label that appears the most often,
i.e., if P(fraud|x) > 0.5, predict fraud cost-sensitive loss: the label that minimizes the
“empirical risk”.• If P(fraud|x) * $1000 > $90 or p(fraud|x) > 0.09, predict
fraud MSE or mean square error: predict average
How we look for optimal models? Don’t impose “exact forms”:
Decision Trees, Classification based on Association rules, Production rules
Learner estimate structure as well as parameters
NP-hard for most “model representation”
Impose “exact forms”: logistic regression functions,
linear regression model, etc Learners estimate parameter
ONLY. Structure is pre-fixed Inductive Bias.
Decision tree is rather flexible, efficient yet powerful representation.
Consider Decision Tree Compromise between accuracy and model
complexity We think that simplest-structured hypothesis that fits
the data is the best. We employ all kinds of heuristics to look for it.
info gain, gini index, Kearns-Mansour, etc pruning: MDL pruning, reduced error-pruning, cost-
based pruning. Reality: tractable, but still pretty expensive Truth: none of purity check functions guarantee
accuracy over testing data.
Random Decision Tree -classification, regression, probability estimation
Key characteristics: Structure is randomly picked. Statistics are summarized from training data.
At each node, an un-used feature is chosen randomly A discrete feature is un-used if it has never
been chosen previously on a given decision path starting from the root to the current node.
A continuous feature can be chosen multiple times on the same decision path, but each time a different threshold value is chosen
Continued We stop when one of the following
happens: A node becomes too small. Or the total height of the tree exceeds some
limits:• Such as the total number of features.
Node Statistics Classification and Probability
Estimation: Each node of the tree keeps the number of
examples belonging to each class.
Regression: Each node of the tree keeps the mean value of
examples sorted into the node
Classification/Prob Estimatimation
During classification, each tree outputs posterior probability:
B1 < 0.5
Y
B2 > 0.7 B1 > 0.3
P1: 200P2: 10
N
Y N
P1: 30P2: 70
Y
… …
P(P1|x)=0.3
Regression During classification, each tree average
value of training examples that falls within each node
Age >30
Y
Capt> 70% Edu=PhD
AvgAGI=100K
N
Y N
AvgAGI=150K
Y
… …
Classification
The prediction from multiple random trees are averaged as the final output.
Classification: loss function is needed.
A few words about some of its advantage Training can be very efficient.
Particularly true for very large datasets.
Natural multi-class probability. Natural multi-label classification and
probability estimation. Imposes very little about the
structures of the model.
Number of trees Sampling theory:
The random decision tree can be thought as sampling from a large (infinite when continuous features exist) population of trees.
Unless the data is highly skewed, 30 to 50 gives pretty good estimate with reasonably small variance. In most cases, 10 are usually enough.
Worst scenario Only one feature is relevant. All the rest are noise. Probability:
Variance Deduction:
Donation Dataset-classification and prob estimation
Decide whom to send charity solicitation letter.
It costs $0.68 to send a letter. Loss function
Result
Credit Card Fraud-classification and prob estimation
Detect if a transaction is a fraud There is an overhead to detect a
fraud, {$60, $70, $80, $90} Loss Function
Result
Comparing with Boosting Don’t handle multi-class problems
naturally, ECOC Do not output probabilities. Inefficient. Boosting rounds is tricky. Sometimes,
more rounds can lead to overfitting. Inefficient. Implementation needs careful numerical
manipulation.
Comparing with Bagging Could be very inefficient particularly
for very large dataset i.e., bootstrap sampling needs linear
scan of the data. Do not output reliable probabilities.
Probability Estimation
Probability Estimation
Overfitting
Non-overfitting of RDT
Selectivity
Tolerance to data insufficiency
GUIDE
Age >30
Y
Capt> 70% Edu=PhD
MLR
N
Y N
MLR
Y
… …
MLR y = a+a1*x1+a2*x2 + … ak*xk
Regression: single independent variable
RDT
Depend on combination of 5 independent variables
RDT
It grows like …
Comparing with GUIDE Need to decide grouping variables and
independent variables. A non-trivial task. If all variables are categorical, GUIDE
becomes a single CART regression tree. Strong assumption and greedy-based
search. Sometimes, can lead to very unexpected results, like the one given earlier
Conclusion Imposing a particular form of model is not
a good idea to train highly-accurate models.
It may not even be efficient for some forms of models.
RDT has been show to solve all three major problems in data mining, classification, probability estimation and regressions, simply, efficiently and accurately.
Selected Bibliography of RDT ICDM’03: “Is random model better? On its accuracy and efficiency”
(Fan, Wang, Yu and Ma) AAAI’04: “On the Optimality of Posterior Probability Estimation by
Random Decision Tree” (Fan) ICDM’05: “Effective Estimation of Posterior Probabilities: Explaining
the Accuracy of Randomized Decision Tree Approaches” (Fan, Greengrass, McCloskey, Yu, and Drummey)
ICDM’05: “Learning through Changes: An Empirical Study of Dynamic Behaviors of Probability Estimation Trees” (Zhang, Buckles, Peng, and Xu)
Master Thesis by Tony Liu, supervised by Kai Ming Ting, “The Utility of Randomness in Decision Tree Construction”, Monash University, 2005
KDD’06: “A General Framework for Fast and Accurate Regression by Data Summarization in Random Decision Trees”