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Mining High-Speed Data
Streams
Pedro Domingos
Geoff Hulten
Sixth ACM SIGKDD International Conference - 2000
Presented by: Tyler J. Sawyer
UVM Spring 2014 - CS 332 Data Mining
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Outline
→ Introduction
→ Hoeffding Trees
→ The VFDT System
→ Performance Study
→ Conclusion / Summary
→ Review Questions
3
Outline
→ Introduction
→ Hoeffding Trees
→ The VFDT System
→ Performance Study
→ Conclusion / Summary
→ Review Questions
4
Introduction
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In today’s society, the ability to extract and interpret knowledge and
data quickly and efficiently is an increasingly important task.
Many organizations today have expandable databases that grow at a
rate of several million records per day.
Mining these databases yield the following:
o Unique opportunities for data analysis
o Complex challenges to overcome
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Introduction - Cont.
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Knowledge Discovery Systems are limited by the following:
o Time
o Memory
o Sample Size
Traditional Systems:
o Amount of available data is small
o Systems use a fraction of their computation power to avoid overfitting
Current Systems:
o Bottleneck is time and memory
o Majority of sample data is unused; underfitting issues surface.
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Introduction - Cont.
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Today’s Algorithms:
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Efficient, but cannot handle supermassive
databases.
Current Data Mining systems are not equipped to
handle the exponential increase of data expansion
New examples arrive at a higher rate than they can
be mined
→ Data Corruption!
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Introduction - Cont.
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Requirements for ‘Modern’ Algorithms:
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Operate continuously and indefinitely
Incorporate new examples as they become available
Never lose potentially valuable information
Build a model using at most one scan of a database or dataset
Use only a fixed amount of main memory.
Require small, constant time per record.
Make a usable model that can be available at any point during
the algorithm’s runtime.
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Introduction - Cont.
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What can fulfill these requirements?
Incremental Learning Methods
 Online Methods
 Successive Methods
 Sequential Methods
While these methods are efficient, they are not always
accurate.
These methods rarely recover from a set of unfavorable
early examples
o
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Outline
→ Introduction
→ Hoeffding Trees
→ The VFDT System
→ Performance Study
→ Conclusion / Summary
→ Review Questions
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Hoeffding Trees
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Classic Decision Tree Learners
o Examples: ID3, C4.5, CART
o Assumes examples can be stored simultaneously in main
memory; loss of learnable examples.
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Disk-based Decision Tree Learners
o Examples: SLIQ, SPRINT
o Assumes examples are stored on disk.
o Big Datasets easily fill disk and errors occur when the dataset
is too large to fit.
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Hoeffding Trees - Cont.
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A typical type of Classification Problem
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Given : N training examples in the form (x,y)
y = discrete class label
x = vector of d attributes
Goal: Produce a model, y = f(x), to predict classes y
of future examples x with high accuracy.
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Hoeffding Trees - Cont.
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Challenge : Design a decision tree learner for
extremely large (potentially infinite) datasets with high
accuracy and low computational cost.
Given a stream of examples:
o The first ones will be used to choose the root test
o Succeeding ones will pass to corresponding leaves
o Pick the best attributes at each leaf
o Continue process recursively
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Hoeffding Trees - Cont.
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But how do we decide how many examples
are necessary at each node?
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Use a statistical result!
the Hoeffding bound (Chernoff bound)
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Hoeffding Trees - Cont.
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Hoeffding Bound :
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G: heuristic measure used to choose test attributes
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C4.5 ⇒ information gain
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CART ⇒ Gini index
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Assume G(.) is to be maximized
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G: heuristic measure after seeing n examples
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Xa: attribute with the highest observed G
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Xb: second-best attribute
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△G: difference between Xa and Xb
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△G = G(Xa) - G(Xb) > 0
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δ: probability of choosing the wrong attribute
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Hoeffding Trees - Cont.
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The Hoeffding Bound:
o after n examples, If △G > ϵ
 Xa is the best attribute with probability 1 - δ
Node needs to accumulate examples from the stream until ϵ
becomes smaller than △G.
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R = range of a real numbered random variables, r
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n = independent observations of this variable.
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Hoeffding Tree Algorithm
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Inputs:
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S : sequence of examples
X : set of discrete attributes
G(.) : split evaluation function
δ : desired probability of choosing the wrong attribute at any
given node
Output:
o HT : A decision tree (Hoeffding Tree)
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Hoeffding Tree Algorithm - Cont.
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Hoeffding Tree Algorithm - Cont.
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Hoeffding Tree Algorithm - Cont.
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Hoeffding Trees - Cont.
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Hoeffding Tree Algorithm guarantees under realistic assumptions the trees
generated will be similar to batch learners.
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p1 : Leaf Probability (assume this is a constant)
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HTδ : Tree produced by HT algorithm with desired δ given an infinite sequence
of examples, S.
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DT* : Decision tree produced by choosing at each node the attribute with the
best G.
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△i : Intentional disagreement between two decision trees.
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P(x) : Probability that the attribute vector x will be observed.
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l(x) : indicator function (1 : True, 0 : False)
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⇒ △i (DT1, DT2) = Σx P(x) l [Path1(x) ≠ Path2(x)]
Theorem 1:
E[△i (HTδ,DT*)] < δ / p
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Hoeffding Trees - Cont.
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Suppose Xa and Xb differ by roughly 10%.
According to
o δ = 0.1% requires only 380 examples
o δ = 0.0001% requires only 345 more examples.
An exponential improvement in δ can be obtained
with a linear increase in the number of examples.
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Outline
→ Introduction
→ Hoeffding Trees
→ The VFDT System
→ Performance Study
→ Conclusion / Summary
→ Review Questions
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The VFDT System
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Very Fast Decision Tree learner (VFDT)
A decision tree learning system
Based on the Hoeffding Tree algorithm
VFDT allows the use of either information gain or the
Gini index as the attribute evaluation measure.
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The VFDT System - Cont.
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Includes a number of refinements to the
Hoeffding Tree algorithm:
Ties
o G-Computation
o Memory
o Poor Attributes
o Initialization
o Rescans
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The VFDT System - Ties
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Two or more attributes may have similar G’s
A large number of examples may be required to decide
between them with high confidence.
In this case, the chosen attribute makes little difference.
In a VFDT, we specify a user-threshold, τ
Thus, if △G < ϵ < τ : split on current best attribute.
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The VFDT System - G-Computation
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The most significant part of the time cost per example is
recomputing G.
Computing a G value for every new example is
inefficient.
In a VFDT, users can specify an nmin value.
nmin : Number of new examples that must accumulate at
a leaf before recomputing G.
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The VFDT System - Memory
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a VFDT’s memory use is dominated by the memory
required to keep counts for all growing leaves.
If the maximum available memory is reached, VFDT
deactivates the least promising leaves.
The least promising leaves are considered to be the
ones with the lowest values of plel.
When a leaf is deactivated, its memory is freed, except
for a single number used to store the value of plel.
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The VFDT System - Poor Attributes
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a VFDT’s memory usage is also minimized by dropping
early on attributes that do not look promising.
As soon as the difference between an attribute’s G and
the best one’s becomes greater than ϵ, then the
attribute can be dropped.
The memory used to store the corresponding counts
can also be freed.
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The VFDT System - Initialization
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VFDT can be initialized with the tree produced by a
conventional RAM-based learner on a small subset of
the data.
The tree can either be input as it is or over-pruned.
Gives VFDT a “head start”
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The VFDT System - Rescans
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VFDT can rescan previously-seen examples.
Rescans are activated if:
o The data arrives slowly enough that time allows for rescans
o The dataset is finite and small enough that it is feasible
VFDT will never grow a tree smaller than ones
produced by other algorithms.
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Outline
→ Introduction
→ Hoeffding Trees
→ The VFDT System
→ Performance Study
→ Conclusion / Summary
→ Review Questions
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Synthetic Data Study
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Comparing VFDT with C4.5 Release 8
Restricted Two Systems to using the same amount of RAM
VFDT used information gain as the G function.
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14 concepts were used, all with 2 classes and 100 attributes.
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For each level after the first 3:
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A fraction f of all the nodes were replaced by leaves
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The rest became splits on a random attribute.
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At depth of 18, all the nodes were replaced with leaves.
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Each leaf was randomly assigned a class.
Stream of training examples were then generated
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Sampling uniformly from the instance space.
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Assigning classes according to the target tree.
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Various levels of class and attribute noise was added.
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Synthetic Data Study - Cont.
Accuracy as a function of the number of training examples
δ = 10-7
nmin = 200
τ = 5%
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Synthetic Data Study - Cont.
Tree Size as a function of the number of training examples
δ = 10-7
nmin = 200
τ = 5%
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Synthetic Data Study - Cont.
Accuracy as a function of the noise level
C4.5 : 100k examples, VFDT: 20 million examples
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Lesion Study
Effect of Initializing VFDT with C4.5 with and without pruning
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Web Data - Trial Run
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Application of VFDT to mine the stream of Web Page Requests
Test Location : The Entire University of Washington Campus
δ = 10-7, nmin = 200, τ = 5%
Statistics for mining 1.6 million examples:
o VFDT took 1450 seconds to do one pass over the training data
o 983 seconds were spent reading data from the disk
o C4.5 took 24 hours to mine 1.6 million examples.
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Web Data - Trial Run Results
VFDT Performance on Web Data
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Outline
→ Introduction
→ Hoeffding Trees
→ The VFDT System
→ Performance Study
→ Conclusion / Summary
→ Review Questions
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Conclusion - Hoeffding Trees
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A method for learning online
Learns from the increasingly common high-volume data
streams
Allows learning in very small constant time per example
Strong guarantees of high asymptotic similarities to
corresponding batch trees.
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Conclusion - VFDT Systems
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A high-performance data mining system
Based on Hoeffding trees
Empirical studies show its effectiveness in taking
advantage of massive numbers of examples
Practical, efficient, and accurate.
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Outline
→ Introduction
→ Hoeffding Trees
→ The VFDT System
→ Performance Study
→ Conclusion / Summary
→ Review Questions
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Review Questions - 1 of 3
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Question: Name four challenges that modern
algorithms have to overcome today.
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Answer: See Slide 7.
Operate continuously and indefinitely
Incorporate new examples as they become available
Never lose potentially valuable information
Build a model using at most one scan of a database or dataset
Use only a fixed amount of main memory.
Require small, constant time per record.
Make a usable model that can be available at any point during the
algorithm’s runtime.
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Review Questions - 2 of 3
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Question: List the input requirements of the HTAlgorithm, and state what output is generated.
o Answer: See Slide 16
o Inputs:
 S : sequence of examples
 X : set of discrete attributes
 G(.) : split evaluation function
 δ : desired probability of choosing the wrong attribute at any given
node
o Output:
 HT : A decision tree (Hoeffding Tree)
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Review Questions - 3 of 3
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Question: How is memory management handled
differently in a VFDT than a Hoeffding Tree?
o Answer: See Slide 27 (& 28).
o VFDT’s memory use is dominated by the memory required to keep
counts for all growing leaves.
o If the maximum available memory is reached, VFDT deactivates the
least promising leaves.
o The least promising leaves are considered to be the ones with the
lowest values of plel.
o When a leaf is deactivated, its memory is freed, except for a single
number used to store the value of plel.
o Might also state early-on attributes are dropped for memory efficiency
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Any Questions?