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 1.Introduction 2.Objective 3.Drawbacks of Previous work 4.Related Work 5.Architecture for Approximate Query Processing 6.Classical Sampling Techniques 7.Special Case of a Fixed Load 8.Lifting Workload to Query Distributions 9.Relational for Stratified Sampling 10.Solution for Single-Table Selection Queries with Aggregation 11.Extensions for General Work Load 12.Comparisons 13.Experimental Results 14.Summary 15.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

Architecture

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 y i while approximate answer is y i ’ Squared error in answering a GROUP BY query Q : SE(Q) = (1/g) Σ i ((y i – y i ’)/ y i )²  Given a probability distribution of queries p w Mean squared error for the distribution: MSE(p w ) = Σ Q p w (Q)*SE(Q), (where p w (Q) is probability of query Q) Root mean squared error (L 2 ): RMSE(p w ) = √ MSE(p w )  Other error metrics L 1 metric : 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 ] = S 2 /k; (d) the variance in estimating Y is E[(μ·n−Y ) 2 ] = n 2 S 2 /k; and (e) the relative squared error in estimating Y is E[((μ·n − Y )/Y ) 2 ] = n 2 S 2 /Y 2 k.

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/ n 2 ∑ j n j 2 S j 2 / k j ; (d) the variance in estimating Y is E[(μ· n−Y ) 2 ] = ∑ j n j 2 S j 2 / k j ; and (e) the relative squared errorin estimating Y is E[((μ · n − Y )/Y ) 2 ] = 1/ Y 2 ∑ j n j 2 S j 2 /k j.

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 R 1, R 2, …, R r such that for any region R j, each query in W selects either all records in R j 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 P w : lifted workload, probability distribution P w (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) R Q, 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. n2 or n4 large (large overlap), P {Q} (R’) is high n1 or n3 large (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 S 2. 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 ] = S 2 /k.

Stratified Sampling (Cont… ) Product ID Revenue 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 p w. k 1, … k r are unknown variables such that Σk j = k. From Equation (2) on earlier slide, MSE(p W ) can be expressed as a weighted sum of the MSE of each query in the workload: Lemma 2: MSE(p W ) = Σ i w i MSE(p{Q})

Allocation (cont…) For any Q ε W, we express MSE(p {Q} ) as a function of the k j ’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(p w ) as a function of the k j ’s variables. Corollary 1 : MSE(p w ) = Σ j (α j / k j ), where each α j is a function of n 1,…,n r, δ, and γ. α j captures the “importance” of a region; it is positively correlated with n j as well as the frequency of queries in the workload that access R j.  Now we can minimize MSE(p w ). Lemma 4: Σ j (α j / k j ) is minimized subject to Σ j k j = k if k j = k * ( sqrt(α j ) / Σ i sqrt(α i ) ) This provides a closed-form and computationally inexpensive solution to the allocation problem since α j depends only on δ, γ and the number of tuples in each fundamental region.

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. Solution for SUM aggregate

Cont… Allocation: Like COUNT, we express an optimization problem with h*r unknowns k 1,…, k h*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 y j (Y j ) be the average (sum) of the aggregate column values of all records in region R j. Since the variance within each region is small, each value within the region can be approximated as simply y j. Thus to express MSE(p {Q} ) as a function of the k j ’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

Putting all together

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 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

Results : Quality vs Sampling Fraction

Cont…

Quality vs Overlap between Training Set and Test Set

Quality vs Data Skew

Cont…

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 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