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Machine and Statistical Learning for Database Querying Chao Wang Data Mining Research Lab Dept. of Computer Science & Engineering The Ohio State University Advisor: Prof. Srinivasan Parthasarathy Supported by: NSF Career Award IIS-0347662 Copyright 2006, Data Mining Research Lab Outline • Introduction – Selectivity estimation – Probabilistic graphical model • Querying transaction database • Probabilistic model-based itemset summarization • Querying XML database • Conclusion Copyright 2006, Data Mining Research Lab Introduction Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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 … Copyright 2006, Data Mining Research Lab Directed Graphical Models (Bayesian Network) x2 x4 x1 x6 x3 x5 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) Copyright 2006, Data Mining Research Lab Undirected Graphical Models (Markov Random Field (MRF)) x2 x4 x1 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) Copyright 2006, Data Mining Research Lab Inference – Computing Conditional Probabilities • Conditioning x2 x4 x1 x6 x3 x5 • Marginalization: • Conditional probabilities Copyright 2006, Data Mining Research Lab Querying Transaction Database Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Frequent Itemset Mining • Market-Basket Analysis 1 A B C D 0 1 1 0 Copyright 2006, Data Mining Research Lab Frequent Itemset Mining • Support(I): number of transactions “containing I” 1 1 1 1 1 1 1 1 1 1 1 Copyright 2006, Data Mining Research Lab Frequent Itemset Mining Problem • Given D, minsup Find all itemsets I with support(I) ≥ minsup Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab An ME Distribution Example Frequent Itemsets X1 • The maximum entropy distribution has the following product form: X2 X3 X4 X5 X1 X2 X1 X3 X2 X3 X3 X4 X4 X5 Where I(.) is an indication function for the corresponding itemset constraint and the constants u0, u1, …, u11 are estimated from the data. X1 X2 X3 Copyright 2006, Data Mining Research Lab An MRF Example X1 C1 X2 X3 C2 X4 C3 X5 Copyright 2006, Data Mining Research Lab Iterative Scaling Algorithm • Time complexity Runs for k iterations, m itemset constraints and t is the average inference time O(k * M * t) Efficient inference is crucial ! Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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) ? Copyright 2006, Data Mining Research Lab Graph Partitioning-Based Approximate MRF Learning Lemma: 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. Copyright 2006, Data Mining Research Lab 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 variablecluster and combine all local models to derive an approximate global MRF Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Accumulative Edge Weighting Scheme • Edge weight should reflect the correlation strength 3+2= Itemsets Support X1 X2 3 X1 X3 4 X2 X3 2 X3 X4 2 X4 X5 6 X1 X2 X3 2 Copyright 2006, Data Mining Research Lab 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? Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Treewidth Based Augmentation Variable-cluster 1-hop neighboring nodes 2-hop neighboring nodes …… Copyright 2006, Data Mining Research Lab Interaction Importance and Treewidth Based Variable-Cluster Augmentation Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Learning Algorithm Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab A Greedy Inference Algorithm X1X2X3X6X7 X3X4X6X8 X5X9X10 M1 M2 M3 Qx = X1 X2 X3 X4 X5 P( X 1, X 2, X 3) P( X 3, X 4) P( X 5) P(Qx ) P( X 3) Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Results on the Web Data • Support threshold = 20, results in 9901 frequent itemsets • Treewidth = 28 according to Maximum Cardinality Search (MCS)-ordering heuristic Copyright 2006, Data Mining Research Lab Varying k (g = 5): Online Time Estimation accuracy Online time Copyright 2006, Data Mining Research Lab Offline Time Varying g (k = 20): Online Time Estimation Accuracy Online time Copyright 2006, Data Mining Research Lab Offline Time Varying tw (k = 25): Online Time Estimation Accuracy Copyright 2006, Data Mining Research Lab Offline Time 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 Copyright 2006, Data Mining Research Lab Probabilistic Model-Based Itemset Summarization Copyright 2006, Data Mining Research Lab 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 nonredundant information Copyright 2006, Data Mining Research Lab The Inclusion-Exclusion Principle Copyright 2006, Data Mining Research Lab Deduction Rules via InclusionExclusion • Let A, B, C, … be items • Let A’ correspond to the set { transactions t | t contains A } • (AB)’ = (A)’ ∩ (B)’ • Then supp(AB) = | (AB)’| Copyright 2006, Data Mining Research Lab Deduction Rules via InclusionExclusion • 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 Copyright 2006, Data Mining Research Lab 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 sABC ≤ sAB + sAC + sBC - sA - sB - sC + n 3 Copyright 2006, Data Mining Research Lab 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 = U We know Supp(J) exactly without counting! Copyright 2006, Data Mining Research Lab 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), IJ} Copyright 2006, Data Mining Research Lab 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 nonderivable itemsets • Can we do better? – Further compress the patterns Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Itemset Summarization Algorithm Copyright 2006, Data Mining Research Lab Generalized Non-Derivable Itemsets • All the itemsets in the final summary are non-derivable • Relax the requirement for an itemset to be derivable Copyright 2006, Data Mining Research Lab Experimental Results • Experimental Setup – Datasets: – Performance metrics: • Summarization accuracy (restoration error) • Summary size • Summarizing time Copyright 2006, Data Mining Research Lab Results on the Chess Dataset minSup = 2000 Summary size 166581 frequent itemsets 1276 non-derivable Summarizing time Estimation accuracy Copyright 2006, Data Mining Research Lab Results on the Chess Dataset Skewed itemset distribution when varying error threshold Copyright 2006, Data Mining Research Lab Results on the Mushroom Dataset minSup = 2031 (25%) Summary size 5545 frequent itemsets 534 non-derivable Summarizing time Estimation accuracy Copyright 2006, Data Mining Research Lab Results on the Mushroom Dataset Skewed itemset distribution when varying error threshold Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Query XML Database – Exploiting Independence Structure from Complex Structural Patterns Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Querying XML Database • An XML document example: DBLP.xml (Digital Bibliography & Library Project) Copyright 2006, Data Mining Research Lab Querying XML Database • A twig example: FOR all books IN document(“DBLP.xml") WHERE publisher = "Morgan Kaufmann" RETURN title b b: book p: publisher p t t : title Copyright 2006, Data Mining Research Lab Querying XML Database b selectivity = 2 p Copyright 2006, Data Mining Research Lab b: book t p: publisher t : title 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 Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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? Copyright 2006, Data Mining Research Lab Estimation Procedure Lemma: If these two tree augmentations are conditionally independent (conditioned on T), then we have: T e1 e2 x y T1 Augmenting T with e1 to get T1 : selectivity T2 Augmenting T with e2 to get T2 Copyright 2006, Data Mining Research Lab Decomposition Strategies • How to decompose a large twig into smaller sub-twigs? – Recursive decomposition with or without voting – Fixed-sized decomposition – Hybrid decomposition Copyright 2006, Data Mining Research Lab Recursive Decomposition a b d e c a b d e a b d f g f g f g c a b d e f a b d a b d f g Recursively applying the estimation formula. f g c a b d f 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 Copyright 2006, Data Mining Research Lab Fixed-sized Decomposition a b c a b c d d f e g b + c d e Very fast, but can not be applied directly b b + d f + d e Copyright 2006, Data Mining Research Lab f g Hybrid Decomposition a b c fixed-sized decomposition a b c d a a b b d b a d b c f e g b c d + e …… c b + d a b b d f + d f g e recursive decomposition with voting b Copyright 2006, Data Mining Research Lab Summary Statistics • What to store in lattice summary? – Store important statistics – Store non-redundant information – How to achieve this? • Store non-derivable patterns only! Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab 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 | estim( q) count( q) | | W | q W count( q) Copyright 2006, Data Mining Research Lab Accuracy of Estimators Avg. Rel Error(%) Recursive Decomp+Voting Fast Decomp Recursive Decomp TreeSketches 20 18 16 14 12 10 8 6 4 2 0 4 NASA 5 6 7 Query Size 8 9 • Recursive decomposition with voting yields best estimates • The quality of estimation degrades as the twig size increases due to error propagation Copyright 2006, Data Mining Research Lab Varying Summary Size TreeLattice TreeSketches Avg. Rel Error(%) 14% 12% 10% 8% 6% 4% 2% 0% 10k NASA 20k 30k 40k 50k Summary Size • The larger the summary, the better the estimations • TreeLattice makes more efficient use of the memory budget Copyright 2006, Data Mining Research Lab Estimation Time Response Time(ms) Recursive Decomp+Voting Fast Decomp Recursive Decomp TreeSketches 70 60 50 40 30 20 10 0 4 NASA 5 6 7 Query Size 8 9 • 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. Copyright 2006, Data Mining Research Lab 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? Copyright 2006, Data Mining Research Lab δ-derivable Pruning TreeSketches IMDB • Larger δ is good for large twigs, at the cost of sacrificing estimation accuracy for small twigs. Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab Conclusion Copyright 2006, Data Mining Research Lab 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 Copyright 2006, Data Mining Research Lab