from Time-Changing Data Streams

LEARNING REGRESSION TREES

Blaž SovdatAugust 27, 2014

)

Data arrives in the form of examples (tuples) Examples arrive sequentially, one by one No control over the speed and order of arrival The underlying “process” that generates stream examples

might change (non-stationary data) Use a limited amount of memory, independent of the

size of the stream (infinite data)

THE STREAM MODEL

(adult, female, 3.141, 0.577)(child, male, 2.1728, 0.1123)(child, female, 2.1728, 1.12)(child, male, 149, 1.23)

…

Example

Requirements of the data stream environment:

1) Process one example at a time, inspect it only once

2) Use a limited amount of memory

3) Work in a limited amount of time

4) Be ready to predict at any time Typical use of data stream learner:

a) The learner receives a new example from the stream (1)

b) The learner processes the example (2, 3)

c) The learner is ready for the next example (4) Different evaluation techniques

DATA STREAM ENVIRONMENT

Data stream prediction cycle

Alfred Bifet and Richard Kirkby. Data Stream Mining: A Practical Approach. 2009.

Regression tree represents a mapping from attribute space to real numbers

Examples are tuples of attribute values Each attribute has a range of possible values:

1) Discrete (also “categorial”) attribute: sex with range {male, female}

2) Numeric attribute: temperature with range R (reals) The target attribute is a real number Concrete example:

INTERMEZZO: DECISION TREES

Example

Example: ((male,first,adult),no)

Tom Mitchell. Machine Learning. McGraw Hill. 1997.

Famous batch learner for regression trees Start with a set of examples , i.e., the training set Each example is of the form , where Pick the attribute that maximizes standard deviation

reduction (SDR) Partition the set according to the attribute Recursively apply the procedure on each subset

INTERMEZZO: CART

𝑠𝑑 (𝑆 )=√ 1

¿𝑆∨¿ ∑(𝒙 , 𝑦 ) ∊ 𝑆

(𝑦−𝑦 )2  ¿𝑠𝑑𝑟 ( 𝐴 )=𝑠𝑑 ( 𝑆 )−∑

𝑖=1

𝑑

¿ 𝑆𝑖∨¿

¿𝑆∨¿ 𝑠𝑑(𝑆𝑖)¿¿

𝑦=1

¿𝑆∨¿ ∑(𝒙 ,𝑦 )∊ 𝑆

𝑦 ¿𝑆 𝑖= {(𝒙 , 𝑦 )∈𝑆|𝐴 (𝒙 )=𝑎𝑖 }

𝐴=argmax𝐴

𝑠𝑑𝑟 (𝐴)

L. Breiman, J. Friedman, C.J. Slone, R.A. Olshen. Classification and Regression Trees. CRC Press. 1984.

Let’s modify CART to a streaming setting Data is not available in advance, and we only see a (small)

sample of the stream When and on what attribute to split? What attribute is “the best” relative to the whole stream? Idea: Apply Hoeffding bound to confidently decide

when to split

A PROBLEM

Well-known result from probability, also known as additive Chernoff bound, proved by Wassily Hoeffding

Many applications in theoretical computer science (randomized algorithms, etc.) and machine learning (PAC bounds , Hoeffding trees, etc.)

Theorem (Hoeffding, 1963). Let be a sum of independent bounded random variables, with , and let . Then

The result is independent of the distribution

SIMPLIFIED HOEFFDING BOUND

Randomized Quicksort does at most comparisons “with high probability”

Rajeev Motwani, Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press. 1995.

Wassily Hoeffding. Probability Inequalities for Sums of Bounded Random Variables. Journal of the American Statistical Association. 1963.

Let and be the best and the second-best attributes (i.e. attributes with highest SDRs)

Let and be estimated standard deviation reductions, computed from examples, for attributes and

If , then with probability at least , where

To see this, solve for Note that means is better than , i.e., it is obvious that

iff , assuming SDRs are positive This is all we need to scale up the CART learner:

Each leaf accumulates examples until it is confident it found the “truly best” attribute

APPLYING THE HOEFFDING BOUND

Elena Ikonomovska. Algorithms for Learning Regression Trees and Ensembles on Evolving Data Streams. PhD thesis. 2012.

Learning:

1) Start with an empty leaf (the root node)

2) Sort a newly arrived example into a leaf

3) Update statistics, compute SDRs, and compute

4) Accumulate examples in the leaf until

a) Split the leaf: create new leaf nodes Predicting:

1) Sort example down the tree, into a leaf

2) Predict the average of examples from the leaf

FAST INCREMENTAL MODEL TREES

The big picture

ε=√ log ( 1𝛿 )2𝑛

Elena Ikonomovska. Algorithms for Learning Regression Trees and Ensembles on Evolving Data Streams. PhD thesis. 2012.

Handling numeric attributes (histogram, BST, etc.) Stopping criteria (tree size, thresholds, etc.) Fitting a linear model in leaves (unthresholded

perceptron) Handling concept drift (with Page-Hinkley test)

EXTENSIONS OF THE FIMT LEARNER

Syntactically no difference between regression and classification (almost) A variant of the FIMT-DD learner available in QMiner The learner exposed via QMiner Javascript API Pass algorithm parameters and data stream specification in JSON format Several stopping and splitting criteria Change detection mechanism, using Page-Hinkley test Can export the model anytime (XML and DOT formats supported) Usage examples available on GitHub The algorithm expects two (learning) or three (predicting) parameters:

1) vector of discrete attribute values;

2) vector of numeric attribute values;

3) target variable value (not needed for prediction)

REGRESSION TREES IN QMINER

REGRESSION TREES IN QMINER

// algorithm parameters var algorithmParams = { "gracePeriod": 300, "splitConfidence": 1e-6, "tieBreaking": 0.005, "driftCheck": 1000, "windowSize": 100000, "conceptDriftP": false, "maxNodes": 15, "regLeafModel": "mean" "sdrThreshold": 0.1, "sdThreshold": 0.01, "phAlpha": 0.005, "phLambda": 50.0, "phInit": 100,};

// describe the data stream var streamConfig = { "dataFormat": ["A", "B", "Y"], "A": { "type": "discrete", "values": ["t", "f"] }, "B": { "type": "discrete", "values": ["t", "f"] }, "Y": { "type": "numeric" }};

// create a new learner var ht = analytics.newHoeffdingTree(streamConfig, algorithmParams);

// process the stream while (!streamData.eof) { /* parse example */ ht.process(vec_discrete, vec_numeric, target);}// use the model var val = ht.predict(["t", "f"], []);

// export the model ht.exportModel({ "file": "./sandbox/ht/model.gv", "type": "DOT" });

The algorithm is pretty fast: tens of thousands of examples per second

Scales poorly with the number of attributes (quadratic in )

When using information gain as attribute selection criterion, needs time

Numeric attribute discretization is expensive (both space and time)

Would love to get feedback from people From now on: Change the algorithm as needed

REGRESSION TREES IN QMINER

Been flirting with NIPS 2013 paper A completely different approach to regression tree

learning Essentially boils down to approximate nearest

neighbor search Very general setting (metric-measure spaces) Strong theoretical guarantees

THE END

Samory Kpotufe, Francesco Orabona. Regression-tree Tuning in a Streaming Setting. NIPS 2013.