machine and statistical learning for database querying

Copyright 2006, Data Mining Research Lab

Machine and Statistical Learning for Database Querying

Chao WangData Mining Research Lab

Dept. of Computer Science & EngineeringThe Ohio State University

Advisor: Prof. Srinivasan Parthasarathy

Supported by: NSF Career Award IIS-0347662


Outline

• Introduction– Selectivity estimation– Probabilistic graphical model

• Querying transaction database

• Probabilistic model-based itemset summarization

• Querying XML database

• Conclusion


Introduction


Introduction

• Database querying

• Selectivity estimation– Estimation of a query result size in database

systems– Usage: for query optimizer to choose an

efficient execution plan

• Rely on probabilistic graphical models


Probabilistic Graphical Models

• Marriage of graph theory and probability theory

• Special cases of the basic algorithms discovered in many (dis)guises:– Statistical physics– Hidden Markov models– Genetics– Statistics– …

• Numerous applications – Bioinformatics – Speech– Vision, – Robotics, – Optimization– …


p(x1,x2,x3,x4,x5,x6) = p(x1)p(x2|x1) p(x3|x1)p(x4|x2)p(x5|x3)p(x6|x2,x5)

Directed Graphical Models (Bayesian Network)

x1

x2x4

x6

x3 x5


p(x1,x2,x3,x4,x5,x6) = (1/Z)Φ(x1,x2) Φ(x1,x3)Φ(x2,x4)Φ(x3,x5)Φ(x2,x5,x6)

Undirected Graphical Models (Markov Random Field (MRF))

x1

x2x4

x3 x5

x6


Inference – Computing Conditional Probabilities

x1

x2x4

x3 x5

x6

• Conditioning

• Marginalization:

• Conditional probabilities


Querying Transaction Database


Transaction Database

• Consist of records of interactions among entities

• Two examples:– Market-basket data

Each basket is a transaction consisting of items

– Co-authorship data

Each paper is a transaction consisting of “author” items


Querying Transaction Database

• Rely on frequent itemsets to learn graphical models

• Rely on the model to solve the selectivity estimation problem– Given a conjunctive query Q, estimate the size

of the answer set, i.e., how many transactions satisfy Q


Frequent Itemset Mining

• Market-Basket Analysis

A B C D

1 0 1 1 0


Frequent Itemset Mining

• Support(I): number of transactions “containing I”

1 11 1

1

1 1

1 1

1 1


Frequent Itemset Mining Problem

• Given D, minsup

Find all itemsets I with support(I) ≥ minsup


Using Frequent Itemsets to Learn an MRF

• A k-itemset can be viewed as a constraint on the underlying distribution generating the data

• Given a set of itemsets, we compute a distribution satisfying them and having a Maximum Entropy (ME)

• This maximum entropy distribution is equivalent to an MRF


An ME Distribution Example

Frequent Itemsets

X1

X2

X3

X4

X5

X1 X2

X1 X3

X2 X3

X3 X4

X4 X5

X1 X2 X3

• The maximum entropy distribution has the following product form:

Where I(.) is an indication function for the corresponding itemset constraint and the constants u0, u1, …, u11 are estimated from the data.


An MRF Example

X1

X2 X3

X4

X5

C1

C2

C3


Iterative Scaling Algorithm

• Time complexityRuns for k iterations, m itemset constraints

and t is the average inference time

O(k * M * t)

Efficient inference is crucial !


Junction Tree Algorithm

• Exact inference algorithm

• Time complexity is exponential in the treewidth (tw) of the model– Treewidth = (maximum clique size in the

graph formed by triangulating the model – 1)

• Real world models, tw is often well above 20, thus intractable


Approximate Inference Algorithm• Gibbs sampling

– Simulating samples from posterior distributions– Sum over samples to evaluate marginal probabilities

• Mean field algorithm– Convert the inference problem to an optimization problem, and

solve the relaxed optimization problem• Loopy belief propagation

– Apply Pearl’s belief propagation directly to loopy graphs– Works quite well in practice

Will the iterative scaling algorithm still converge (when subjected to

approximate inference algorithms) ?


Graph Partitioning-Based Approximate MRF Learning

For all disjoint vertex subsets a, b and c in an MRF, whenever b and c are separated by a in the graph, then the variables associated with b, c are independent given the variables associated with a alone.

Lemma:


Graph Partitioning-Based Approximate MRF Learning

• Cluster variables based on graph partitioning

• Interaction importance and treewidth based variable-cluster augmentation

• Learn an exact local MRF on a variable-cluster and combine all local models to derive an approximate global MRF


Clustering Variables

• k-MinCut – Partition the graph into k equal parts – Minimize the number of edges of E whose

incident vertices belong to different partitions – Weighted graphs: Minimize the sum of

weights of all edges across different partitions


Accumulative Edge Weighting Scheme

Itemsets SupportX1 X2 3X1 X3 4X2 X3 2X3 X4 2X4 X5 6

X1 X2 X3 2

3+2=

• Edge weight should reflect the correlation strength


Clustering Variables

• The k-MinCut partitioning scheme yields disjoint partitions. However, there exist edges across different partitions. In other words, different partitions are correlated to each other. So how do we account for the correlations across different partitions?


Interaction Importance and Treewidth Based Variable-Cluster Augmentation

• Augmenting variable-cluster– Add back most significant incident edges to a

variable-cluster

• Optimization– Take into consideration model complexity

• Keep track of treewidth of the augmented variable-clusters• 1-hop neighboring nodes first, then 2-hop nodes, …, and so

on


Treewidth Based Augmentation

Variable-cluster

1-hop neighboring nodes

2-hop neighboring nodes

… …


Interaction Importance and Treewidth Based Variable-Cluster

Augmentation


Approximate Global MRFs

• For each augmented variable-cluster, collect related itemsets and learn an exact local MRF

• All local MRFs together offer an approximate global MRF


Learning Algorithm


A Greedy Inference Algorithm

• Given the global model consisting of a set of local MRFs, how do we make inference?– Case 1: all query variables are covered by a single

MRF, evaluate the marginal probability directly– Case 2: use a greedy decomposition scheme to

compute• First, pick a local model that has the largest intersection with

the current query (i.e., cover most variables)• Then pick the next local model covering most uncovered

query variables, and so on• Overlapped decomposition


A Greedy Inference Algorithm

Qx = X1 X2 X3 X4 X5

X1X2X3X6X7 X3X4X6X8 X5X9X10

M1 M2 M3

1, 2, 3 3 4 5

3

( ) ( , ) ( )( )

( )x

P X X X P X X P XP Q

P X


Discussions

• The greedy inference scheme is a heuristic• Global model is not globally consistent;

However, we expect that the global model is nearly consistent ( Heckerman et al. 2000)

• A generalized belief propagation style approach is currently under investigation to force the local consistency across the local models, thereby offering a globally consistent model


Experimental Results

• C++ implementation. The Junction tree algorithm is implemented based on Intel’s Open-Source Probabilistic Networks library (C++)

• Use Apriori algorithm to collect frequent itemsets

• Use Metis for graph partitioning


Experimental Setup• Datasets

– Microsoft Anonymous Web, |D|=32711, |I|=294– BMS-Webview1, |D|=59602, |I|=497

• Query workloads– Conjunctive queries, e.g., X1 & ¬X2 & X4

• Performance metrics– Time: online estimating time and offline learning time– Error: average absolute relative error

• Varying – k, the no. of clusters– g, the no. of vertices used during the augmentation– tw, the treewidth threshold when using treewidth based augmentation

optimization


Results on the Web Data

• Support threshold = 20, results in 9901 frequent itemsets

• Treewidth = 28 according to Maximum Cardinality Search (MCS)-ordering heuristic


Varying k (g = 5):

Estimation accuracy

Online time

Online Time

Offline Time


Varying g (k = 20):

Estimation Accuracy

Online time

Online Time

Offline Time


Estimation Accuracy

Online Time

Offline Time

Varying tw (k = 25):


Using Non-Redundant Itemsets

• There exist redundancies in a collection of frequent itemsets

• Select non-redundant patterns to learn probabilistic models

• Closely related to pattern summarization


Probabilistic Model-Based Itemset Summarization


Non-Derivable Itemsets

• Based on redundancies– How do supports relate?

• What information about unknown supports can we derive from known supports?– Concise representation: only store non-

redundant information


The Inclusion-Exclusion Principle


Deduction Rules via Inclusion-Exclusion

• Let A, B, C, … be items

• Let A’ correspond to the set{ transactions t | t contains A }

• (AB)’ = (A)’ ∩ (B)’

• Then supp(AB) = | (AB)’|


Deduction Rules via Inclusion-Exclusion

• Inclusion-exclusion principle:|A’ U B’ U C’| = |A’| + |B’| + |C’|

- |(AB)’| - |(AC)’| - |(BC)’|

+ |(ABC)’|

Thus, since |A’ U B’ U C’| ≤ n,

Supp(ABC) ≤ s(AB) + s(AC) + s(BC)

- s(A) - s(B) - s(C) + n


Complete Set for Supp(ABC)0 sABC ≥ 0

1 sABC ≤ sAB

sABC ≤ sAC

sABC ≤ sBC

2 sABC ≥ sAB + sAC - sA

sABC ≥ sAB + sBC – sB

sABC ≥ sAC + sBC – sC

3 sABC ≤ sAB + sAC + sBC - sA - sB - sC + n


Derivable Itemsets

Given: Supp(I) for all I J

Lower bound on Supp(J) = L

Upper bound on Supp(J) = U

• Without counting: Supp(J) [L, U]

• J is a derivable itemset (DI) iff L = UWe know Supp(J) exactly without counting!


Derivable Itemsets

• J is a derivable itemset:– No need to count Supp(J)– No need to store Supp(J)

• We can use the deduction rules

– Concise representation:C = { (J, Supp(J) ) | J not derivable from Supp(I),

I J }


Probabilistic Model Based Itemset Summarization

• We can learn the MRF from non-derivable itemsets alone

Lemma: Given a transaction dataset D, the MRF M constructed from all of its σ-frequent itemsets is equivalent to M’, the MRF constructed from only its σ-frequent non-derivable itemsets

• Can we do better?– Further compress the patterns


Probabilistic Model Based Itemset Summarization

• Use smaller itemsets to learn an MRF

• Use this model to infer the supports of larger itemsets

• Use those itemsets whose occurrence can not be explained (by some error threshold) by the model to augment the model


Itemset Summarization Algorithm


Generalized Non-Derivable Itemsets

• All the itemsets in the final summary are non-derivable

• Relax the requirement for an itemset to be derivable


Experimental Results

• Experimental Setup– Datasets:

– Performance metrics:• Summarization accuracy (restoration error)

• Summary size

• Summarizing time


Results on the Chess Dataset

Estimation accuracy

Summary size

Summarizing time

minSup = 2000

166581 frequent itemsets

1276 non-derivable


Results on the Chess Dataset

Skewed itemset distribution when varying error threshold


Results on the Mushroom Dataset

Estimation accuracy

Summary size

Summarizing time

minSup = 2031 (25%)

5545 frequent itemsets

534 non-derivable


Results on the Mushroom Dataset

Skewed itemset distribution when varying error threshold


Result Summary and Discussions

• There do exist redundancies in a collection of itemsets, and the probabilistic model based summarization scheme can effectively eliminate such redundancies– When datasets are dense and largely satisfy conditional

independence assumption, our summarization approach is extremely efficient

– When datasets become sparse and do not satisfy the conditional independence assumption, the summarization task becomes more difficult (need more time and space)

• Itemsets-based MRF learning and MRF-based itemset summarization are two interactive procedures


Query XML Database – Exploiting Independence Structure from Complex Structural Patterns


Querying XML Database• XML is becoming the standard for data exchange• We need to query the structure and text data of XML

documents• XML twig query:

– an important query mechanism

– a structural query with small branches

• Optimizing these queries requires estimating the selectivity of the twig queries


Querying XML Database

• An XML document example: DBLP.xml

(Digital Bibliography & Library Project)



• A twig example:FOR all books IN document(“DBLP.xml")WHERE publisher = "Morgan Kaufmann"RETURN title

b

p t

b: book

p: publisher

t : title



b

p t

b: book

p: publisher

t : title

selectivity = 2


Problem Statement

• The goal is to accurately estimate the selectivity of twig queries with limited memory– Need a structure to store relevant statistics of

the data– Then estimate selectivity from these statistics


Our Approach (TreeLattice)

• Key idea: store the occurrence statistics of small twigs in the summary– The summary is a lattice consisting of small

trees, thus called TreeLattice

• Then based on these statistics to estimate the selectivity of the large twigs


Challenges

• How to estimate the selectivity for a given twig given the selectivity information of its sub-twigs?

• How to decompose a large twig into smaller twigs?• What statistics to store in the lattice summary?


Estimation Procedure

T

y

e2

x

e1

T1 T2

x

Augmenting T with e1 to get T1

y

Augmenting T with e2 to get T2

Lemma: If these two tree augmentations are conditionally independent (conditioned on T), then we have:

: selectivity


Decomposition Strategies

• How to decompose a large twig into smaller sub-twigs?– Recursive decomposition with or without

voting – Fixed-sized decomposition– Hybrid decomposition


Recursive Decompositionab

c d fe g

abd fe g

ab

c d fg

abd f

g

abd fe

abd f

g

ab

c d f

Recursively applying the estimation formula.

It’s possible there exist multiple feasible decompositions. Rely on voting to obtain the best estimate as we can

• Much more accurate than without voting• Estimating process slows down


Fixed-sized Decomposition

a

b

c d

a

b

c d f

e g

a

b

c d

a

b

c d

b

c d

e

b

c d

e

b

c d

e

+b

d f

e

+b

d f

g+

b

d f

e

b

d f

e

b

d f

g

b

d f

g

Very fast, but can not be applied directly


Hybrid Decomposition

a

b

c d f

e g

… …recursive

decomposition with voting

a

b

d

a

b

c

b

d

a

b

b

c

a

b

a

b

c d

b

c d

e

+ b

d f

e

+b

d f

g+

fixed-sized decomposition


Summary Statistics

• What to store in lattice summary?– Store important statistics – Store non-redundant information– How to achieve this?

• Store non-derivable patterns only!


Summary Statistics

• A twig pattern is δ- derivable if and only if its true selectivity is within an error tolerance of δ to its expected selectivity according to TreeLattice.– 0-derivable (δ=0) patterns are those patterns whose

selectivity can be estimated exactly.

• Pruning 0-derivable patterns – No loss of accuracy


Summary Statistics

• Level-wise lattice summary construction– Add all twigs of size 1&2 to the summary (base)– Then add larger non-derivable frequent twigs

into the summary, until the memory budget is depleted


Experimental Methodology

• Datasets: NASA, PSD, IMDB and XMark

• Workloads: 1000 frequent twig queries of size between 4 and 9.

• Error metric: Mean absolute relative error

1

|W |

| estim(q) count(q) |count(q)qW


Accuracy of Estimators

02468

101214161820

4 5 6 7 8 9Query Si ze

Avg.

Rel

Err

or(%

)

Recursi ve Decomp+Voti ng Recursi ve DecompFast Decomp TreeSketches

NASA

• Recursive decomposition with voting yields best estimates

• The quality of estimation degrades as the twig size increases due to error propagation


Varying Summary Size

0%

2%

4%

6%

8%

10%

12%

14%

10k 20k 30k 40k 50k

Summary Si ze

Avg.

Rel

Err

or(%

)

TreeLatti ce TreeSketches

NASA

• The larger the summary, the better the estimations

• TreeLattice makes more efficient use of the memory budget


Estimation Time

010203040506070

4 5 6 7 8 9Query Si ze

Resp

onse

Tim

e(ms

)

Recursi ve Decomp+Voti ng Recursi ve DecompFast Decomp TreeSketches

NASA

• TreeLattice is very fast when processing relative small twigs

• Recursive decomposition with voting slows down a lot as the twig size increases.

• Overall, fast decomposition is best.


δ-derivable Pruning

• The proportion of 0-derivable patterns is very high on NASA, PSD and XMark– Tree growing conditional independence

assumption holds well– TreeLattice works very well

• Assumption does not hold that well on IMDB. How to improve the estimations on IMDB?


δ-derivable Pruning

• Larger δ is good for large twigs, at the cost of sacrificing estimation accuracy for small twigs.

IMDB

TreeSketches


Discussions

• TreeLattice is effective in estimating the selectivity of XML twig queries– Compares favorably with the state-of-the-art

approach– The lattice summary construction is fast– The online estimation is fast


Conclusion


Conclusion

• Conditional independence structure is common in the real world

• Graphical models are effective to capture such structures and solve the selectivity estimation problem for database querying

• Model structured data (sequence/tree/graph) using probabilistic models

• Model streaming/incremental data

machine and statistical learning for database querying

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