LOCAL STRUCTURE AND DETERMINISM IN PROBABILISTIC DATABASES. Theodoros Rekatsinas, Amol Deshpande, Lise Getoor
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1 LOCAL STRUCTURE AND DETERMINISM IN PROBABILISTIC DATABASES Theodoros Rekatsinas, Amol Deshpande, Lise Getoor
2 Motivation Probabilistic databases store, manage and query uncertain data Numerous applications generate correlated data (e.g., sensor networks, data integration etc.) Problem Scaling query evaluation in probabilistic databases with correlated data
3 Contributions in a nutshell Challenges in correlated TPC-H probabilistic Query B3 + A DBs: 10 1 Our 1. How to represent correlations? Approach λ a1 Baselines λ 0 Annotated Arithmetic a0 (red, blue) Circuit (AAC) Evaluation Time (sec) Exploits global and local structure of distribution AAC - indep. DB 2. How to evaluate queries VE efficiently - cor. DB? Online merging λ 10 b * * + B * * B VE - indep. DB VE(ADD) - indep. DB λ b1 0.4 Transform queries into AACs; merge with DB AACs 0 * VE(ADD) - cor. DB AAC - cor. DB TPC-H Scale factor
4 Outline Background Arithmetic Circuits in probabilistic DBs Experiments Conclusion and future work
5 Outline Background Arithmetic Circuits in probabilistic DBs Experiments Conclusion and future work
6 Example correlated probabilistic DB S A B s1 m 1 T B C t1 1 p s2 n 1 Query Query: q(c):- s(a,b),t(b,c) Lineage: l = (s1 t1) (s2 t1) f1 Database i1 f2 r1 f3 s1 s2 t1 s1 s1 f1 f 1 (s1) Factor graph Query Evaluation Probabilistic inference Inference has exponential complexity i2 s2 f2 s2 s2 t1 f 3 (s2, t1) t1 f3 f 2 (s2)
7 Speeding inference Compilation: separate inference into two phases Offline: compile distribution into a data structure Online: use data structure to answer queries Goal: Push work to offline phase to optimize online inference time Exploit local and global structure Global structure: Model topology (treewidth) Local structure: Exhibited in model parameters Determinism Context specific independence (CSI)
8 Determinism Example Bayesian network [Darwiche, UAI `06] Battery Age Alternator Fan Belt Battery Battery Power Charge Delivered Starter Battery Power Fuel OK Pump WEAK Gas DEAD Lights ON OFF Fuel Line Distributor Gas Gauge If Battery Power = Dead, then Lights = OFF Radio Lights Engine Turn Over Engine Start Spark Plugs
9 Context specific independence Example Bayesian network [Darwiche, UAI `06] Battery Age Alternator Fan Belt Battery Charge Delivered Fuel Pump Fuel Line Starter Gas Distributor Battery Power Gas Gauge Spark Plugs Radio Lights Engine Turn Over Engine Start
10 Context specific independence Example Bayesian network [Darwiche, UAI `06] Battery Age Alternator Fan Belt Battery Charge Delivered Fuel Pump Fuel Line Starter Gas Distributor Battery Power Gas Gauge Spark Plugs Radio Lights Engine Turn Over Engine Start
11 Overview of correlation representations Data correlations factor graphs [Sen et al., `07] pc-tables [Green et al., `06] junction trees [Kanagal et al., `10] - compilation - exploit global structure arithmetic circuits [Darwiche, `03], AACs [this paper] - compilation - exploit global and local structure Query correlation augmented factor graphs (variable elimination) lineage, OBDDs [Olteanu et al., `08], D-trees [Olteanu et al., `10] - OBDDs, D-trees exploit local strucure lineage [this paper]
12 Example arithmetic circuit Factor graph A B MLF (a 0 = false, a 1 = true): λ a0 λ b0 θ a0 θ b0 a 0 + λ a0 λ b1 θ a0 θ b1 a 0 + λ a1 λ b0 θ a1 θ b0 a 1 + λ a1 λ b1 θ a1 θ b1 a 1 f a f ab A f a (.) A B f ab (.) factorization + * * λ a * * λ a λ b0 0.7 λ b1
13 Querying an arithmetic circuit Arithmetic circuit + λ a0 0.6 * * + * * λ a1 0.4 Probability of evidence query Pr(A = false, B = false) Set λ a0 = 1, λ a1 = 0, λ b0 = 1, λ b1 = 0 Evaluate the circuit to get Pr(A = false, B = false) = λ b0 0.7 λ b1
14 Outline Background Arithmetic Circuits in probabilistic DBs Experiments Conclusion and future work
15 Overview of our Framework Offline Problem Pre-processing No online updates DB Factor Graph Data Compile DB Arithmetic Circuits Query Lineage Formulas Online Query Evaluation Merge Arithmetic Circuits Convert to Annotated Arithmetic Circuits Solution (bad) Repeat entire compilation process (expensive) Solution (good) Result Tuple Annotated Arithmetic Circuits (AACs): Add variable annotations to summation Arithmetic nodes, Circuit value annotations to edges Merge AACs during query evaluation For each result tuple Final AC Evaluate Probability
16 Overview of our Framework Offline Pre-processing Online Query Evaluation DB Factor Graph Data Query Lineage Formulas Convert to Annotated Arithmetic Circuits Compile DB Annotated Arithmetic Circuits Result Tuple Annotated Arithmetic Circuit Merge Annotated Arithmetic Circuits For each result tuple Final AC Evaluate Probability
17 Example Annotated Arithmetic Circuit A B f ab f ab A f a (.) A B f ab (.) λ a A 0 1 * * + B * * + B * λ b λ b1 λ a1 0.4
18 Overview of our Framework Offline Pre-processing Online Query Evaluation DB Factor Graph Data Query Lineage Formulas Convert to Annotated Arithmetic Circuits Compile DB Annotated Arithmetic Circuits Result Tuple Annotated Arithmetic Circuit Merge Annotated Arithmetic Circuits For each result tuple Final AC Evaluate Probability
19 Compiling Factor Graphs Extend compilation technique based on VE [Chavira et al., `07] Uses Algebraic Decision Diagrams (ADDs) Add variable annotations to summation nodes VE requires a variable ordering Output AACs follow the reverse VE ordering Collection of disjoint ordered sets of variables One for each disconnected part of the factor graph
20 Example Database S A B s1 m 1 s2 n 1 Factor graph s1 s2 f1 f2 T B C t1 1 p t1 f3 Compile + s * * λ s VE Ordering: O = s 1, s 2, t 1 Partial Orderings: O 1 = s 1, O 2 = s 2, t λ s11 λ t10 λ s * + t 1 0 * * + s * λ t11 λ s t 1 *
21 Overview of our Framework Offline Pre-processing Online Query Evaluation DB Factor Graph Data Query Lineage Formulas Convert to Annotated Arithmetic Circuits Compile DB Annotated Arithmetic Circuits Result Tuple Annotated Arithmetic Circuit Merge Annotated Arithmetic Circuits For each result tuple Final AC Evaluate Probability
22 Compiling Lineage Use Ordered Binary Decision Diagrams [Olteanu et al., 2008] Lineage consists of only deterministic (AND/OR) factors More efficient compilation Compilation algorithm (sketch) 1. Choose a total variable ordering 2. Compile lineage formula to an OBDD 3. Construct an AAC from an OBDD
23 Getting a total ordering OBDD construction requires a total variable ordering Finding an optimal variable ordering is NP-hard Designing a heuristic (ideas): 1. Connected variables in lineage must appear together [Ebendt et al., 2005] 2. More influential variables must appear sooner in the ordering [Ebendt et al., 2005] 3. Total ordering must respect partial ordering constraints Necessary for merging 4. Variables in a constraint must be kept together Facilitates caching during merging
24 Variable Ordering under Partial Order Constraints Algorithm (Goal: Find total order of variables) 1. Score variables by their position in the partial orderings 2. Group random variables that are connected in the lineage 3. Use scores to get a total variable ordering
25 Overview of our Framework Offline Pre-processing Online Query Evaluation DB Factor Graph Data Query Lineage Formulas Convert to Annotated Arithmetic Circuits Compile DB Annotated Arithmetic Circuits Result Tuple Annotated Arithmetic Circuit Merge Annotated Arithmetic Circuits For each result tuple Final AC Evaluate Probability
26 Merging Annotated Arithmetic Circuits Algorithm: Traverse the query and DB AACs simultaneously At each step merge exactly two AACs (rooted at summation nodes) If the result can be computed immediately, done Else: choose one root and recurse on its sub-circuits Choose the root with the variable that appears first in the total ordering Complexity: # O% m $ i A s i & ( ' m: size of lineage-aac, A: set of relevant database AACs, s i : size of i-th AAC in A
27 Outline Background Arithmetic Circuits in probabilistic DBs Experiments Conclusion and future work
28 Experimental Evaluation Experimental evaluation using the TPC-H benchmark Tuple independence Schema level correlations Multiple scale factors (0.001, 0.005, 0.01, 0.05, 0.1) Evaluation on queries with both safe and not-safe evaluation plans Boolean versions of TPC-H queries Q2, Q3, Q5, Q6, Q8 and Q16 without top-level aggregates Comparison to: Variable elimination with Conditional Probability Tables [Sen et al., `07] Variable elimination with Algebraic Decision Diagrams Captures determinism and context specific independence
29 Compilation Time 10 3 Database Compilation Time Compilation Time (sec) Independent DB Correlated DB TPC-H Scale factor
30 Evaluation Time 10 1 TPC-H Query B TPC-H Query B8 Evaluation Time (sec) VE - indep. DB VE(ADD) - indep. DB AAC - indep. DB VE - cor. DB VE(ADD) - cor. DB AAC - cor. DB Evaluation Time (sec) VE - indep. DB VE(ADD) - indep. DB AAC - indep. DB VE - cor. DB VE(ADD) - cor. DB AAC - cor. DB TPC-H Scale factor TPC-H Scale factor Missing points: Evaluation time exceeded 100 sec
31 Evaluation Time Longer lineage formulas TPC-H Query B TPC-H Query B16 Evaluation Time (sec) VE - indep. DB VE(ADD) - indep. DB AAC - indep. DB VE - cor. DB VE(ADD) - cor. DB AAC - cor. DB TPC-H Scale factor Evaluation Time (sec) Missing points: Evaluation time exceeded 100 sec VE - indep. DB VE(ADD) - indep. DB AAC - indep. DB VE - cor. DB VE(ADD) - cor. DB AAC - cor. DB TPC-H Scale factor
32 Micro-benchmarking TPC-H Query B3 evaluation breakdown TPC-H Query B16 evaluation breakdown Evaluation Time (sec) Merging Phase Lineage-to-AAC Variable Ordering Lineage Creation Evaluation Time (sec) Merging Phase Lineage-to-AAC Variable Ordering Lineage Creation Not optimized (blue) TPC-H Scale factor TPC-H Scale factor Longer lineage formula
33 Outline Background Arithmetic Circuits in probabilistic DBs Experiments Conclusion and future work
34 Conclusion and future work Conclusion Exploit compilation, global and local structure for efficient exact query evaluation At least one order of magnitude improvement Introduced Annotated Arithmetic Circuits New merging algorithm to support updates Of independent interest to probabilistic reasoning Open problems Lifted Inference in Arithmetic Circuits isomorphic parts of an arithmetic circuit with respect to operations and θ variables? Approximate inference in Arithmetic Circuits How to incorporate approximations into the framework? Thank you!
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