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A Robust, Optimization-Based Approach for Approximate Answering of Aggregate Queries. Surajit Chaudhuri Gautam Das Vivek Narasayya Presented By: Vivek Tanneeru Venkata Dinesh Jammula. Outline. Introduction Objective Drawbacks of Previous work Related Work
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A Robust, Optimization-Based Approach for Approximate Answering of Aggregate Queries Surajit Chaudhuri Gautam Das Vivek Narasayya Presented By: Vivek Tanneeru Venkata Dinesh Jammula
Outline • Introduction • Objective • Drawbacks of Previous work • Related Work • Architecture for Approximate Query Processing • Classical Sampling Techniques • Special Case of a Fixed Load • Lifting Workload to Query Distributions • Relational for Stratified Sampling • Solution for Single-Table Selection Queries with Aggregation • Extensions for General Work Load • Comparisons • Experimental Results • Summary • References
1. Introduction • Decision Support applications - OLAP and data mining for analyzing large databases • Approximate answers to queries given accurately and efficiently benefit the scalability of these applications • Workload information in picking samples of the data
2. Objective • Pre-compute a sample as an optimization problem • Minimize error in estimation of aggregates • Implemented on Microsoft SQL Server 2000, for an effective solution to be deployed in Commercial DBMS
3. Drawbacks of Previous work • Lack of rigorous problem formulations lead to solutions that are difficult to evaluate theoretically • Does not deal with uncertainty in expected workload • Ignores the variance in data distribution of aggregated columns
4. Related Work • Weighted Sampling • Outlier Index • Congressional Sampling • On the fly Sampling • Histograms
5. Architecture for Approximate Query Processing Preliminaries: • Consider Queries with selections, foreign-key joins and GROUP BY, containing aggregation functions such as COUNT, SUM and AVG. • Assume a pre-designated amount of storage space is available for selecting samples from the database • Selecting samples can be randomized or deterministic
Error Metrics • If correct answer for query Q is y while approximate answer is y’ Relative error : E(Q) = |y - y’| / y Squared error : SE(Q) = (|y - y’| / y)² • If correct answer for the ith group is yi while approximate answer is yi’ Squared error in answering a GROUP BY query Q : SE(Q) = (1/g) Σi((yi – yi’)/ yi)² • Given a probability distribution of queries pw • Mean squared error for the distribution: MSE(pw) =ΣQpw(Q)*SE(Q), (where pw(Q) is probability of query Q) • Root mean squared error (L2): RMSE(pw) = √MSE(pw) • Other error metrics • L1metric : the expected relative error over all queries in workload • L∞ metric : the max error over all queries
6. Classical Sampling Techniques Uniform Sampling: LEMMA 1 (a) μ is an unbiased estimator for y, namely, E[μ] = y; (b) μ· n is an unbiased estimator for Y namely E[μ· n] = Y ; (c) the variance (or standard error) in estimating y is E[(μ− y) 2] = S2/k; (d) the variance in estimating Y is E[(μ·n−Y ) 2] = n2S2/k; and (e) the relative squared error in estimating Y is E[((μ·n − Y )/Y ) 2] = n2S2/Y2k.
Classical Sampling Techniques Stratified Sampling: LEMMA 2 (a) μ is an unbiased estimator for y, namely, E[μ] = y; (b) μ · n is an unbiased estimator for Y, namely, E[μ · n] = Y ; (c) the variance in estimating y is E[(μ − y) 2] = 1/ n2 ∑j nj2 Sj2/ kj ; (d) the variance in estimating Y is E[(μ· n−Y ) 2] = ∑ j nj2 Sj2 / kj ; and (e) the relative squared errorin estimating Y is E[((μ · n − Y )/Y ) 2] = 1/ Y2 ∑ j nj2 Sj2 /kj .
Classical Sampling Techniques Neyman Allocation: LEMMA 3 Given a population R = {y1, . . . , yn}, k and r, the optimal way to form r strata and allocate k samples among all strata is to first sort R and select strata boundaries so that ∑ j n j S j is minimized, and then, for the j th strata, to set the number of samples k j as k j = k(n j S j /∑ j n j S j )
Classical Sampling Techniques • Multivariate Stratified Sampling • Weighted Sampling • Error Estimation and Confidence Intervals
7. Special Case: Fixed Workload • Problem: FIXEDSAMP Input: R, W, k Output: A sample of k records (with appropriate additional columns) such that MSE(W) is minimized.
Fundamental Regions • Fundamental Regions: For a given relation R and workload W, consider partitioning the records in R into a minimum number of regions R1, R2, …, Rr such that for any region Rj, each query in W selects either all records in Rj or none.
Solution for FIXEDSAMP Step 1. Identify Fundamental Regions • Case A. r <= k • Case B. r > k Step 2 Pick Sample Records Step 3 Assign values to additional columns
8. Lifting Workload to Query Distributions • Resilient to the situation when incoming query is “similar” but not identical to queries in the workload • Pw : lifted workload, probability distribution • Pw (Q’) : Related to the amount of similarity of Q’ to the workload • Not concerned with syntactic similarity of query expressions
Lifted workload (Cont.) • Two parameters δ (½ ≤ δ ≤1) and γ (0 ≤ γ ≤ ½) define the degree to which the workload “influences” the query distribution. For any given record inside (resp. outside) RQ, the parameter δ (resp. γ) represents the probability that an incoming query will select this record. • P{Q}(R’)is the probability of occurrence of any query that selects exactly the set of records R’.
Lifted workload (Cont.) • n1 , n2, n3, and n4 are the counts of records in the regions. • n2or n4large (large overlap), P{Q}(R’) is high • n1orn3large (small overlap), P{Q}(R’) is low • We elaborate on this issue by analyzing the effects of (four) different • boundary settings of these parameters. • 1. δ → 1 and γ → 0: implies that incoming queries are identical • to workload queries. • 2. δ → 1 and γ → ½: implies that incoming queries are • supersets of workload queries. • 3. δ → ½ and γ → 0: implies that incoming queries are subsets • of workload queries. • 4. δ → ½ and γ → ½: implies that incoming queries are • unrestricted.
9. Rationale for Stratified Sampling Consider a population, i.e. a set of numbers R = {y1,.,yn}. Let the average be y, the sum be Y and the variance be S2. Suppose we uniformly sample k numbers. Let the mean of the sample be μ. The quantity μ is an unbiased estimator for y, i.e. E[μ] = y the variance (i.e., squared error) in estimating y is E[(μ-y) 2] = S2/k.
Stratified Sampling (Cont… ) Query Q1 : SELECT COUNT(*) FROM R WHERE PRODUCTID IN (3,4); Population POPQ1 = {0,0,1,1} Thus, a stratified sampling scheme partitions R into r strata containing n1, ., nr records (where Σnj = n), with k1, …, kr records uniformly sampled from each stratum (where Σkj = k).
10. Solution for single-table selection queries with Aggregation • Stratification a.) How many strata r to partition relation R into, b.) Records from R that belong to each strata • Allocation how to divide k( the number of records available for the sample) into integers k1, …, kr across r strata such that Σkj = k • Sampling uniformly samples kj records from stratum Rj to form the final sample of k records
Solution for COUNT aggregate • Stratification: From Lemma 1. Lemma 1: For a workload W consisting of COUNT queries, the fundamental regions represent an optimal stratification. • Allocation: We want to minimize the error over queries in pw . k1, … kr are unknown variables such that Σkj = k. From Equation (2) on earlier slide, MSE(pW) can be expressed as a weighted sum of the MSE of each query in the workload: Lemma 2: MSE(pW) = Σiwi MSE(p{Q})
Allocation (cont…) For any Q εW, we express MSE(p{Q}) as a function of the kj’s Lemma 3 : For a COUNT query Q in W, Let ApproxMSE(p{Q}) = Then,
Outline of Proof: • Since we have an (approximate) formula for MSE(p{Q}), we can express MSE(pw) as a function of the kj’s variables. Corollary 1 : MSE(pw) = Σj(αj / kj), where each αj is a function of n1,…,nr, δ, and γ. αj captures the “importance” of a region; it is positively correlated with nj as well as the frequency of queries in the workload that access Rj. • Now we can minimize MSE(pw). Lemma 4: Σj (αj / kj) is minimized subject to Σj kj = k if kj = k * ( sqrt(αj) / Σi sqrt(αi) ) This provides a closed-form and computationally inexpensive solution to the allocation problem since αj depends only on δ, γ and the number of tuples in each fundamental region.
Solution for SUM aggregate Stratification: • Bucketing Technique We further divide fundamental regions with large variance into a set of finer regions, each of which has significantly lower internal variance. • Treat each region as strata • From optimal Neyman Allocation Technique, We have: h*r finer strata Good to have a large h, but h is set to value 6.
Cont… Allocation: • Like COUNT, we express an optimization problem with h*r unknowns k1,…, kh*r. • Unlike COUNT, the specific values of the aggregate column in each region (as well as the variance of values in each region) influence MSE(p{Q}). • Let yj(Yj) be the average (sum) of the aggregate column values of all records in region Rj. Since the variance within each region is small, each value within the region can be approximated as simply yj. Thus to express MSE(p{Q}) as a function of the kj’s for a SUM query Q in W:
Pragmatic Issues • Identifying Fundamental Regions • Handling Large Number of Fundamental Regions • Obtaining Integer Solutions • Obtaining an Unbiased Estimator
11. Extensions • GROUP BY • JOIN • Other Extensions
12. Comparisons Weighted Sampling • Records that are accessed more frequently have a greater chance of being included into the sample • Assumes fixed workload Outlier Indexing • Form their own stratum that is sampled in its entirety • Assumes fixed workload
Comparisons (cont…) Congressional Sampling • Allocation of samples between two strata • To minimize MSE,
13. Experimental Results PREVIOUS WORKS: USAMP – uniform random sampling WSAMP – weighted sampling OTLIDX – outlier indexing combined with weighted sampling CONG – Congressional sampling
Experimental Setup • Databases: Used the popular TPC-R benchmark for experiments • Workloads: Generated several workloads over TCP-R schema using an automatic query generation program • Parameters: Varied the parameters like, • Skew of the data • Sampling fraction between 0.1 % - 10 % • Workload size was varied between 25 - 800 queries • Error Metric: Report the average error over all queries in the workload
Training Set vs Test Set • The basic idea is to split the available workload into two sets: • the training workload and • the test workload • Training Set: The workload used to determine the sample • Test Set: The workload used to estimate the error
14. Summary • A comprehensive solution to the problem of identifying samples for approximately answering aggregation queries • Its implementation on a database system • With a novel technique for lifting a workload, we make our solution robust enough to work well even for workloads that are similar but not identical to the given workload. • Handles the problems of data variance, heterogeneous mixes of queries, GROUP BY and foreign-key joins.
15. References • Surajit Chaudhuri, Gautam Das, Vivek Narasayya: A Robust, Optimization-Based Approach for Approximate Answering of Aggregate Queries. SIGMOD Conference 2001. • Surajit Chaudhuri, Gautam Das, Vivek Narasayya. Optimized Stratified Sampling for Approximate Query Processing. ACM Transactions on Database Systems (TODS), 32(2): 9 (2007)
Thank You Questions ? Presented By: Vivek Tanneeru Venkata Jammula
