Detecting Attribute Dependencies from Query Feedback

Transcription

1 Detectng Attrbute Dependences from Query Feedback Peter J. Haas 1, Faban Hueske 2, Volker Markl 1 1 IBM Almaden Research Center 2 Unverstät Ulm VLDB 2007 Peter J. Haas

2 The Problem: Detectng (Parwse) Dependent Attrbutes Example: Color and Year are ndependent f F( Color = red AND Year = 2005 ) = F ( Color = red ) x F (Year = 2005 ) F( Color = blue AND Year = 2007 ) = F (Color = blue ) x F ( Year = 2007 ) etc. F ( P ) = fracton of rows n table that satsfy predcate P Dependence = sgnfcant departure from ndependence Detecton needed for automatc statstcs confguraton n query optmzers Whch multvarate statstcs should we keep? Need to rank the dependences (lmted space budget) Other uses nclude Schema dscovery for data ntegraton Data mnng (dependency dagrams) Root-cause analyss and system montorng Approaches to detecton and rankng: proactve and reactve 2

3 Outlne Prevous approaches Proactve approach: CORDS Reactve approaches: SASH, Correlaton analyzer Our new reactve approach Dependency detecton Handlng ncomplete feedback, nconsstences Rankng Expermental Results 3

4 A Proactve Approach: CORDS [IMH+, SIGMOD 04] Sample the relaton (or vew) and compute a contngency table: Blue Green Red Compute (robust) ch-squared statstc 2 χ 2 Declare dependency f χ > t Both t and sample sze chosen usng ch-squared theory Can rank attrbute pars by mean-square contngency dstance (MSCD) Normalzed ch-squared statstc ( 200 ( 2670 )( 2670 ) 2670) ( )( ) ( O E) = = + E

5 Reactve Approaches Query System Catalog RUNSTATS Profle Optmzer Analyss Plan Executon Est Card Act Card Populaton Feedback Warehouse Result Focus system resources on nterestng attrbutes Complement proactve approaches Can explot DB2 feedback warehouse 5

6 A Spectrum of Reactve Approaches Correlaton Analyzer (CA) [AHL+, VLDB 04] Our Approach SASH [LWV, VLDB 03] Smple and Cheap Sophstcated and Expensve 6

7 Correlaton Analyzer Uses multple observatons (actuals) for each attrbute par O 1 = {(blue,2005): 0.02, (blue): 0.2, (2005): 0.103} O 2 = {(red,2006): 0.07, (red): 0.82, (2006): 0.11} etc. Computes rato for each par and compares to 1!, e.g. [0.9,1.1] O 1 : 0.02 / (0.2 x 0.103) = 0.97 ndependent O 2 : 0.07 / (0.82 x 0.11) = 0.77 dependent Attrbute dependency f two or more observatons look dependent Ranks attrbutes by weghted sum of volatons Problems Ad hoc procedures, wasted nformaton Unstable: depends on amount, orderng of feedback 7

8 Outlne Prevous approaches Proactve approach: CORDS Reactve approaches: SASH, Correlaton analyzer Our new reactve approach Dependency detecton Handlng ncomplete feedback, nconsstences Rankng Expermental Results 8

9 A New Approach to Dependency Dscovery Lke CORDS, but uses ncomplete contngency table wth exact entres Blue Green Red ?? ?? ???? 450? Declare dependency f H M > u (where H M s our new test statstc) Crtcal value u from extenson of classcal ch-squared theory Normalze H M to get rankng metrc 9

10 The H M Statstc Set H M = M x t Qx M = number of rows n table x =(O E ) / E Q s pseudo-nverse of Σ Note: 1 [,j [ # observatons r = rank of Q fαβ = fracton of rows wth ta. = α and tb. = β (1 fα )(1 ) fβ f = j fα fβ 1 f α f j, α = αj, and β βj Σ f j = α 1 fβ f j, α αj, and β = βj fβ 1 f j, α αj, and β βj Propertes: smlar to H M 0 2 χ H M = 0 ff observatons consstent wth ndependence Larger H M u less consstent wth ndependence 10

11 Choosng the Threshhold u Superpopulaton approach Assume A and B generated by truly ndependent mechansm Theorem: Under ths model, for large # of rows, H M has approxmately a χ 2 r dstrbuton Choose u as (1 p) quantle of for small p. Then Prob χ 2 r { } Prob { χ 2 } M r H > u > u = p 11

12 Mssng Feedback Most mportant case: O = { (blue,2005): 0.02, (blue): 0.2, (2005):? } Assume optmzer estmate of (2005) frequency avalable Assume (rough) upper bound on abs(relatve error of estmate) Can obtan from feedback-warehouse records Fll n mssng frequency for (2005) Derve rough bounds on true value: l [ F (2005) [ u Make frequency as ndependent as possble (conservatve) E.g., F (2005) = 0.1 and E = r 1 = 0 Consder ALL observatons wth mssng (2005) frequency Mnmze Σ (E ) 2 (closed-form soluton avalable) 12

13 Handlng Inconsstency Problem: No full multvarate frequency dstrbuton consstent wth feedback Records collected at dfferent tme ponts Inserts/deletes/updates n between feedback observatons Soluton method 1: use tmestamps to resolve conflcts Soluton method 2: lnear programmng Obtan mnmal adjustment of frequences needed for consstency + mn w ( s + s ) s.t. F(blue,2005) + s s = 0.2 color F(2005) + s s = 0.3 F(2005,color) = F(2005) + s, s 0 for all F '(blue,2005) = F(blue,2005) s + s

14 Rankng Attrbute Pars Problem: normalze H M ( = M xt Q x) to le n [0,1] Guaranteed (conservatve) normalzaton Based on Courant-Fscher Mnmax Theorem * 2 * H η = Md x, where d = largest egenvalue of Q M Can be numercally unstable (huge values of ) Heurstc normalzatons H / z M Table Cardnalty Mnmal number of dstnct values Degrees of freedom of ch-squared dstrbuton 2 χ r 0.99 Quantle of ( effectve upper bound) 14

15 Outlne Prevous approaches Proactve approach: CORDS Reactve approaches: SASH, Correlaton analyzer Our new reactve approach Dependency detecton Handlng ncomplete feedback, nconsstences Rankng Expermental Results 15

16 Normalzaton Constants Rankngs relatvely consstent for dfferent z (choce s not too crtcal) Best results: degrees of freedom, quantles ( hgh probablty upper bound) 16

17 Rankng vs Amount of Feedback New method: Correlaton analyzer: 17

18 Dependency Measure vs Amount of Feedback New method: Correlaton analyzer: Dependency measure Dependency measure 18

19 Executon Tme O(n 3 ) theoretcal complexty Subsecond executon tme for up to 250 feedback records Tmes based on prelmnary Java mplementaton 19

20 Obtanng Practcal Executon Tmes Samplng Stable results wth small # of obs. Sub-second response tmes Incremental mantenance of H M = M x t Q x New observaton = add new row + new column to Σ Want to update Q drectly Q = pseudo-nverse of Σ Apply SVD updatng methods Σ As n latent semantc ndexng E.g., foldng-n method O(k 2 ) 20

21 Conclusons Dependence s everywhere! Query feedback s an effectve way to detect dependence Ch-squared extenson to mplement detecton Attrbutes can be n multple tables Effectve rankng methods Practcal solutons for handlng nconsstent or mssng feedback Acceptable performance usng samplng and ncremental mantenance 21

22 Future Work Hgher-level dependences 6000 Worst-case Error Factor Sngle Column 2 Columns 3 Columns Order of CG Statstcs Full ntegraton of proactve and reactve methods Cf. Aboulnaga et al. [VLDB 2004] 22

23 The End IBM Research My web page: LEO (LEarnng Optmzer) project: research.nsf/pages/r.datamgmt.nnovaton.html 23

24 The End IBM Research Backup Sldes 24

25 The H M Statstc (Based on n Observatons) fαβ fα f β Set x = for = 1,2,, n f f α β f αβ = fracton of rows wth ta. = α and tb. = β Set Σ= Σ j, where (1 fα )(1 ) fβ f = j fα fβ 1 f α f j, α = α j, and β βj Σ f j = α 1 fβ f j, α α j, and β = βj fβ 1 f j, α α j, and β βj 25

26 The H M Statstc, Contnued Symmetrc Shur decomposton: where D = dag( d, d,, d ) 1 2 Set D = dag( d, d,, d ), where d 1 2 1/ d f d > 0 = 0 f d = 0 t Set Q = G DG Set M = # rows n table Then HM = t Mx Qx n n t Σ= GDG Q s pseudo-nverse of Σ: QΣ=Σ Q = I Set r = r( Q) = # postve dagonal entres n D r 26