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Copyright © 2013 Geoffrey I Webb
Fundamental and Advanced Machine Learning Methods for Big Data Applications
Geoffrey I Webb,
Ana Martinez, Nayyar Zaidi, Shenglei Chen
Monash Universityhttp://www.csse.monash.edu.au/~webb
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Copyright © 2013 Geoffrey I Webb
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Copyright © 2013 Geoffrey I Webb
Overview
• Big data
• Classification learning
• Sampling
• Dimensionality reduction
• Scaling-up existing algorithms
• Stream learning
• Bias and variance and big data
• Selective KDB
• Incremental Bayesian Network Classifiers
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Big data
• Can mean many things
– Complex integration of many heterogeneous data sources
– Very large/streaming data
Name (SI
decimal prefixes)
Value Binary usage
kilobyte (kB)
megabyte (MB)
gigabyte (GB)
terabyte (TB)
petabyte (PB)
exabyte (EB)
zettabyte (ZB)
yottabyte (YB)
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
What is ‘big’?
• Number of– instances– dimensions– classes
• Big data usually includes data sets with sizes beyond the ability of commonly used software tools to capture, curate, manage, and process the data within a tolerable elapsed time. Big data sizes are a constantly moving target, as of 2012 ranging from a few dozen terabytes to many petabytes of data in a single data set.– Wikipedia
• Machine learning research usually treats more than 1 million examples as very large.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Examples
• Spelling correction
• Translation
• Farecast
• Recommender systems
• Electoral outcomes
Whitelaw, C, B Hutchinson, GY Chung, & G Ellis. "Using the web for language independent spellchecking and autocorrection." In Proceedings of the 2009 Conference
on Empirical Methods in Natural Language Processing: Volume 2, pp. 890-899. Association for Computational Linguistics, 2009.
Silver, Nate. The Signal and the Noise: Why So Many Predictions Fail-but Some Don't. Penguin Press, 2012.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Not a universal panacea
• Jeopardy but not chess
• Spelling correction and translation but not comprehension
http://www.engadget.com/2011/02/15/watson-soundly-beats-the-humans-in-first-round-of-jeopardy/
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Classification learning
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Evolving distributions
• Key issue
– Is the distribution from which the data are drawn static or dynamic?
– Concept drift
• class membership changes, eg rich
– Concept evolution
• new classes emerge
– Distribution drift
• probabilities change
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Dimension of change
• Normally time but may be other such as location
• Classifier can only take dimension of change into account if data to be classified will fall within current scope or if it is possible to extrapolate
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Training
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Training
Testing
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Loss functions
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Imbalanced classes
• Many big datasets have a rare class of interest and a majority class from which we seek to distinguish it.
– Ad click-through
– Conversions
– Disease
– Fraud
– Homeland security
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Loss functions for imbalanced classes
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Loss functions for imbalanced classes
Predictions
Pos Neg
Act
ual Pos TP FN
Neg FP TN
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Loss functions for imbalanced classes
• Area under the ROC curve
True Positive Rate (TPR)
Predictions
Pos Neg
Act
ual Pos TP FN
Neg FP TN
Prof. William H. Press, “Unit 17: Classifier Performance: ROC, Precision-Recall, and All That.”
http://www.nr.com/CS395T/lectures2008/17-ROCPrecisionRecall.pdf
False Positive Rate (FPR)
Predictions
Pos Neg
Act
ual Pos TP FN
Neg FP TN
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Loss functions for imbalanced classes
• Area under the Precision Recall Curve
Prof. William H. Press, “Unit 17: Classifier Performance: ROC, Precision-Recall, and All That.”
http://www.nr.com/CS395T/lectures2008/17-ROCPrecisionRecall.pdf
Recall = True Positive
Rate (TPR)
Predictions
Pos Neg
Act
ual Pos TP FN
Neg FP TN
Precision
Predictions
Pos Neg
Act
ual Pos TP FN
Neg FP TN
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Mutual information
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Learning curves
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KDB k=2
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Sampling
• Select s instances from a dataset of size n
• Important that sample be selected randomly
• Make sure you use a robust random number generator
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Ideal Sampling
• Select data quantity at which learning curve approaches asymptotic error and learn from sample
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Finding asymptotic error
• Progressive sampling
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Provost, F, D Jensen, T Oates. “Efficient progressive sampling.” In Proc 5th ACM SIGKDD international conference on Knowledge Discovery and Data Mining, pp. 23-32. ACM, 1999.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Hoeffding's bound
Hulten, G, and P Domingos. "Mining complex models from arbitrarily large databases in constant time." In Proceedings 8th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 525-531. ACM, 2002.
Sample size
Error margin
Sample mean
Population mean
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Maximum sample
• Take largest sample capacity can handle
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Data quantity
KDB k=2
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Maximum sample
• Take largest sample capacity can handle
• Saves overheads of repeated sampling and risk of terminating too soon
• Has risk that asymptotic error may not be reached
– but alternative techniques wouldn’t be able to handle a larger sample anyway!
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Sampling with and without replacement
• Sampling involves deciding how many times Ki each element iof a collection should occur in the sample
• Sampling without replacement restricts Ki to 0 or 1
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Uniform fixed-sized sampling with replacement for fixed n
selected ← 0
while selected < s
add a randomly selected instance to the sample
increment selected
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Uniform sequential variable-sized sampling without replacement
i ← 1
while i < n
with fixed probability do
add the next instance to the sample
increment i
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Uniform sequential fixed-sized sampling without replacement for known n
selected ← 0
i ← 1
while selected < s
with probability (s - selected )/(n-i+1) do
add the next instance to the sample
increment selected
increment i
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Uniform sequential fixed-sized sampling without replacement for unknown n
count ← 0
while count < s and count < n
add the next instance to the sample
increment count
while more instances remain
increment count
with probability s/count do
add the next instance to the sample replacing an existing instance selected at random
else
discard the next instance
Tille, Yves. Sampling algorithms. Springer, 2006.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Dimensionality reduction
• Many learning algorithms are super-linear with respect to dimensionality
• Dimensionality can be reduced by
– feature selection
– feature projection
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Feature selection
• Most powerful techniques are too computationally intensive for big data
– Eg wrapper techniques
– Best approach varies depending on base learner
• Techniques that consider only the relationship between an attribute and the class are efficient
– Eg top-k mutual information
– However, overlook complex interactions between attributes
• May be most effective to use powerful technique on a sample
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Feature Projection
• Project feature space onto lower dimensional space
• Principal Components Analysis
• First principal component is the planar projection that maximises variance (= minimises RMSE with respect to original)
• Subsequent principal components are those that maximise variance (= minimise RMSE) while being uncorrelated with prior components
• First few principal components will capture most of the variation (= information) in the data
• Generalisations including principal curves and manifolds project onto manifolds instead of planes
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Scaling-up existing algorithms
• Distributed cloud/cluster computing
• Hadoop
– Commodity clusters
– Map Reduce
• Map problem onto sub-problems and distribute these
• Assemble solution from solutions to sub-problems
White, Tom. Hadoop: The definitive guide. O'Reilly Media, Inc., 2012
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Streaming algorithms
• Handle data that are too large to retain
– computer network/phone traffic, financial transactions, web searches, sensor data
• May be difficult to get labelled data
• Strong memory and running time constraints
– learning rate must be greater than the data rate
– only limited data can be retained
• Real time accuracy evaluation and formalisation, mainly to adjust the parameters accordingly.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Online and incremental learning
• Online learning
– Data arrives as input stream
– Classifier makes prediction
– Then correct classification is revealed and classifier updated
– Examples
• Ad placement, online conversions
• Incremental learning
– Classifier is updated as input arrives
– Classifier is identical to batch classifier
Auer, Peter. “Online Learning.” In Encyclopedia of Machine Learning, C. Sammut and G.I. Webb, Editors. 2010, Springer: New York. p. 736-743.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Streaming Strategies
• Retain samples of data and learn from these
– Continually assess current model against incoming data and when models lose accuracy take new samples and relearn
• Continually update a model using current data
– Refine using new data
– Prune elements that decline in accuracy
• Create ensemble of classifiers each learned from successive time periods
– Retire older classifiers as newer ones are created
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Weighted majority algorithm
• Each classifier E has a weight wtE
• Classification by weighted majority vote
• All incorrect classifiers have their weights reduced wt+1E
=wtE , 0< <1
• Error is bounded to no more than twice the error of the best classifier
Littlestone, N, and MK Warmuth. "The weighted majority algorithm." In 30th Annual Symposium on Foundation of Computer Science, pp. 256-261. IEEE, 1989.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Winnow
Binary attributes
Non-negative real valued weights
Threshold
Prediction Correct xi = 0 xi = 1
1 0 unchanged
0 1 unchanged
Littlestone, N. "Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm." Machine Learning 2(4)(1988): 285-318.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Stochastic gradient descent
• Many classifiers have parameters that are learned by optimisation
e.g. logistic regression and SVM
– usually requires many passes through the data
• For linear classifiers stochastic gradient descent often converges before a single pass is completed.
– global gradient approximated by the gradient at each example
– performs sequential updates
– good step size is essential
• learn from an initial sample
– must take examples in random order
Zhang, Tong. "Solving large scale linear prediction problems using stochastic gradient descent algorithms." In Proceedings 21st International Conference on Machine learning, p. 116. ACM, 2004.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
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Bias and variance
• Learning curves are not all equal
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias and variance
• A major factor in the difference between learning curves
• Decomposition of 0-1 loss
• Bias and variance relate to the performance of the learner given different training sets
“Bias Variance Decomposition.” In Encyclopedia of Machine Learning, C. Sammut and G.I. Webb, Editors. 2010, Springer: New York. p. 100-101.
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias and Variance
1,0,1,1,0,1,0,0,1
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Learner 1,0,1,1,0,1,0,0,?
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Copyright © 2013 Geoffrey I Webb
Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias and Variance
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
Learner 1,0,1,1,0,1,0,0,?
1,1,0,1,0,1,1,1,?1
0
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1
1
0
0
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias and Variance
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
Learner 1,0,1,1,0,1,0,0,?
1,1,0,1,0,1,1,1,?1
0
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1
1
0
0
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias and Variance
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
Learner 1,0,1,1,0,1,0,0,?
1,1,0,1,0,1,1,1,?1
0
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1
1 X
0 X
0
Variance ≈ (lower limit on) error due to variability in response to sampling
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias and Variance
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
Learner 1,0,1,1,0,1,0,0,?
1,1,0,1,0,1,1,1,?1 X
0 X
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1,0,1,1,0,1,0,0,1
1,1,0,1,0,1,1,1,1
0,0,1,1,0,1,0,1,0
1,0,1,1,1,0,1,1,0
1 X
1
0
0 X
Variance ≈ (lower limit on) error due to variability in response to sampling
Bias ≈ error due to central tendency of the learner
Bias = error - variance
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias and variance
High bias Low bias High bias Low bias
High variance High variance Low variance Low variance
Image from Bias Variance Decomposition, in Encyclopedia of Machine Learning, C. Sammut and G.I. Webb, Editors. 2010, Springer: New York. p. 100-101.
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Intrinsic error
• Many bias/variance analyses also include intrinsic error
• For our purposes this is included in bias
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Bias/variance and big data
• As data quantity increases, variance should decrease
• Low variance important for small data
• Low bias important for big data
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Low bias important for big data
• Low bias requires capacity to describe wide variety of multivariate distributions
• Big datasets contain fine detail needed to precisely delineate complex multivariate distributions
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Bias/variance and big data
0
0.1
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0.3
0.4
0.5
0.6
0.7
0.8
0 100,000 200,000 300,000 400,000 500,000 600,000 700,000 800,000 900,000 1,000,000
Ro
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Data quantity
Naïve Bayes
KDB k=2
KDB k=5
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
Most machine learning research has used small data
0
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Data quantity
Naïve Bayes
KDB k=2
KDB k=5
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Computational tractability
• Error will be minimised by low bias algorithms
• Big data require efficient computation
– Linear wrt size
– Learn in a limited number of passes
• Most low-bias learners are compute intensive
– super-linear with respect to data quantity
– Kernel SVM and Random Forests
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k-dependence Bayesian classifier (KDB)
• Bayesian network classifier proposed by Sahami (1995).
• KDB
– the probability of each attribute value is conditioned by the class and at most k other attributes.
– Extends TAN to multiple parents.
C
A1 A2 A3 A4
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Big data | Class learning | Sampling |Dimensionality red’n | Scaling-up | Streams | Bias/variance |Selective KDB | Incremental BNC
k-dependence Bayesian classifier (KDB)
• k=0 is Naïve Bayes
• k variance and bias
• High k with low bias should have low error for big data.
C
A1 A2 A3 A4
0
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0.2
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0 100,000 200,000 300,000 400,000 500,000 600,000 700,000 800,000 900,000 1,000,000
Ro
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ared
Err
or
Data quantity
Naïve Bayes
KDB k=2
KDB k=5
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KDB algorithm
1st pass:• Order attributes according to mutual
information (MI) with the class.
2nd pass:• Assign k parents to each attribute
according to MI conditioned on the class.
• Add the class as parent of all attributes
C
A1 A2 A3 A4
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Two pass learning
No of instances No of attributes No of classesAv no of values/att
No of classes No of attributes Av no of values/att
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Selective KDB - Motivation
• KDB is efficient and effective for large data.
• Irrelevant attributes can increase error.
• Cannot predetermine the best k for a given data quantity.
• Want an efficient way to select attributes and best k.
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Selective KDB
MI(Ai;C)
MI(Ai;Aj,C) A1 A2 A3 A4
C
A1 A2 A3 A4
C
LF1 LF2 LF3 LF4
Leave-one-out cv (Pazzani’s trick)
Attributes ordered by MI
Each alternative model tested is a minor addition to the previous
LF1 LF2 LF3 LF4
A1 A2 A3
C
best
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Selective KDB
• Loss function can be RMSE, 0-1 loss, Matthews Correlation Coefficient (for unbalanced datasets), etc.
• Still the value of k has to be tuned.
– Solution: Selective2 KDB: matrix of loss function results kxn.
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
a1 a2 a3 a4 a5 a6
p1 p1 p1 p1 p1
p2 p2 p2 p2
p3 p3 p3
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Selective KDB
• Loss function can be RMSE, 0-1 loss, Matthews Correlation Coefficient (for unbalanced datasets), etc.
• Still the value of k has to be tuned.
– Solution: Selective2 KDB: matrix of loss function results kxn.
KDB Selective KDB Selective2 KDB
Training time
Test time
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Selective KDB – Results (RMSE)
• Competitive with KDB in 16 very large datasets (165K-54.6M examples):
• Mean best k = 4.11
• Mean % attributes selected = 82.6626.72
KDB
selective KDB 8-8-0 5-11-0 5-11-0 6-10-0 6-9-1
k-selective KDB 5-11-0
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Selective KDB – Results (RMSE)
• Comparison with Random Forest.
• Need to sample in 3/4 (out of 16) datasets to get RF 10/100 results.
RF (5EF) RF (Num)
Trees = 10 Trees = 100 Trees = 10 Trees = 100
k-selective KDB 6-1-6 4-1-7 5-0-8 4-0-8
Mnist
(250K/8.1M)
MITC
(600K/839K)
Satellite
(2M/8.7M)
Splice
(10M/54.6M)
RF (100) Sample 0.29580.0017 0.05180.0007 0.45680.0006 0.05300.0005
k-selective
KDB
Sample 0.23240.0029 0.04550.0019 0.45310.0011 0.05210.0006
All data 0.14490.0007 0.04460.0020 0.44480.0004 0.05230.0002
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Selective KDB – Results (MCC)
• Unbalanced datasets: use MCC as loss function.
• Splice dataset: 0.32% of positive classes.
• Comparison with Random Forest.
KDB selective KDB
0.1768 0.1918
0.1855 0.1984
0.1932 0.2043
0.1986 0.2105
0.2061 0.2148
MITC
(600K/839K)
Splice
(10M/54.6M)
RF (100) Sample 0.9989 0.0950
k-selective
KDB
Sample 0.9954 0.1963
All data 0.9956 0.2148
Discrete attributes
Numeric attributes
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Incremental Bayesian Network Classifiers
x1 x2 x3 xn
y
…
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Incremental naïve Bayes
• Probability estimates are based on counts of the frequency of each attribute value co-occurring with the class
• These can be updated incrementally
• Can these desirable features be generalised to more sophisticated learners?
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Adding edges reduces bias
• With additional edges it is possible to exactly represent all naïve Bayes distributions and more
– Lower bias
– Higher variance
– Should be more accurate for bigger data
– But which edges should we add?
x1 x2 x3 xn
y
…
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Averaged n-Dependence Estimators
• Develop all of a family of classifiers that each add edges to naïve Bayes
• Select order of dependence, n
• Each model selects n attributes
– All other attributes are independent given these attributes and the class
– Each model has lower bias but higher variance than NB
– Ensembling reduces the variance
Webb, GI, JR Boughton, FZheng, KM Ting, HSalem. "Learning by extrapolation from marginal to full-multivariate probability distributions: decreasingly naive Bayesian classification." Machine Learning 86(2) (2012): 233-272
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Averaged n-Dependence Estimators
All subsets of
n attributes
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Averaged n-Dependence Estimators
• Incremental learning in a single pass through the data
• Training time complexity O(man+1)
• Classification time complexity O(an+1k)
• Space complexity O(an+1vn+1k)
Number of
training examples
Number of
attributes
Number
of classes
Average number of
values per attribute
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Averaged n-Dependence Estimators
• As n increases bias decreases
– Good for big data
0
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0 100000 200000 300000 400000 500000 600000 700000 800000 900000 1000000
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Naïve Bayes
A1DE
A2DE
A3DE
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Subsumption resolution
• If P(x1 | x2) = 1.0 then P(y | x1,x2) = P(y | x2)
– Eg P(oedema | female, pregnant) = P(oedema | pregnant)
• Subsumption resolution looks for subsuming attributes at classification time and ignores them
– Simple correction for extreme form of violation of attribute independence assumption
– Very effective in practice – reduce bias at small cost in variance –though not always applicable
– For AnDE with n≥1 uses statistics collected already – no learning overhead – often reduces classification time
Zheng, F, GI Webb, P Suraweera, L Zhu. "Subsumption resolution: an efficient and effective technique for semi-naive Bayesian learning." Machine Learning 87(1)(2012): 93-125.
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Weighting
Jiang, Liangxiao, and Harry Zhang. "Weightily averaged one-dependence estimators." In PRICAI 2006, pp. 970-974. Springer Berlin, 2006.
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Weighting
• Weighting also reduces bias at the cost of a small increase in variance
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Data quantity
A3DE A3DE W
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Weighting and subsumption resolution are complementary
• When SR is applicable, both in combination have lower bias but slightly higher variance than either alone
RMSE
Dataset Size A2DE A2DE-SR A2DE-W A2DE-WSR
small cleveland 303 0.359 0.360 0.361 0.361
balance-scale 625 0.430 0.430 0.430 0.430anneal 898 0.118 0.098 0.116 0.096
large
adult 48,842 0.313 0.306 0.308 0.303localization 164,860 0.499 0.499 0.498 0.498covtype 581,102 0.371 0.349 0.350 0.335poker-hand 1,025,010 0.496 0.496 0.420 0.420kddcup 5,209,460 0.044 0.040 0.043 0.039
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Questions?
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References
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