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z(kx%q{HROKp@D&siG@QzLBRXK^x$OA zQ;Kkr2>}j|=Lv+HK;(hKjS);z-y$64qq4AUaep}L^3UsbamYOh0sgA-$KwNuPRkSieOCUIump^sBO2;WT!NqV%; RqMHE-JYD@<);T3K0RYcNhI#-1 literal 0 HcmV?d00001 diff --git a/slides/talks/2017-1-EDBT-Inference/index.html b/slides/talks/2017-1-EDBT-Inference/index.html index a64e3d88..e7fdeb71 100644 --- a/slides/talks/2017-1-EDBT-Inference/index.html +++ b/slides/talks/2017-1-EDBT-Inference/index.html @@ -79,42 +79,6 @@ -
-
-

Online Aggregation (OLA)

- -

$Avg(3,6,10,9,1,3,9,7,9,4,7,9,2,1,2,4,10,8,9,7) = 6$

-

$Avg(3,6,10,9,1) = 5.8$ $\approx 6$

- -

$Sum\left(\frac{k}{N} Samples\right) \cdot \frac{N}{k} \approx Sum(*)$

- -

Sampling lets you approximate aggregate values with orders of magnitude less data.

-
- -
-

Typical OLA Challenges

-
-
Birthday Paradox
-
$Sample(R) \bowtie Sample(S)$ is likely to be empty.
-
Stratified Sampling
-
It doesn't matter how important they are to the aggregate, rare samples are still rare.
-
Replacement
-
Does the sampling algorithm converge exactly or asymptotically?
-
-
- -
-

Replacement

-
-
Sampling Without Replacement
-
... eventually converges to a precise answer.
-
Sampling With Replacement
-
... doesn't need to track what's been sampled.
-
... produces a better behaved estimate distribution.
-
-
-
-
@@ -214,11 +178,259 @@
-

Key Idea: OLA

+

Idea: Use OLA

+
+
+

Online Aggregation (OLA)

+ +

$Avg(3,6,10,9,1,3,9,7,9,4,7,9,2,1,2,4,10,8,9,7) = 6$

+

$Avg(3,6,10,9,1) = 5.8$ $\approx 6$

+ +

$Sum\left(\frac{k}{N} Samples\right) \cdot \frac{N}{k} \approx Sum(*)$

+ +

Sampling lets you approximate aggregate values with orders of magnitude less data.

+
+ +
+

Typical OLA Challenges

+
+
Birthday Paradox
+
$Sample(R) \bowtie Sample(S)$ is likely to be empty.
+
Stratified Sampling
+
It doesn't matter how important they are to the aggregate, rare samples are still rare.
+
Replacement
+
Does the sampling algorithm converge exactly or asymptotically?
+
+
+ +
+

Replacement

+
+
Sampling Without Replacement
+
... eventually converges to a precise answer.
+
Sampling With Replacement
+
... doesn't need to track what's been sampled.
+
... produces a better behaved estimate distribution.
+
+
+ +
+

OLA over GMs

+
+
Tables are Small
+
Compute, not IO is the bottleneck.
+
Tables are Dense
+
Birthday Paradox and Stratified Sampling irrelevant.
+
Queries have High Tree-Width
+
Intermediate tables are large.
+
+

Classical OLA techniques aren't entirely appropriate.

+
+
+ +
+
+

(Naive) OLA: Cyclic Sampling

+
+ +
+

A Few Quick Insights

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  1. Small Tables make random access to data possible.
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  3. Dense Tables mean we can sample directly from join outputs.
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  5. Cyclic PRNGs like Linear Congruential Generators can be used to generate a randomly ordered, but non-repeating sequence of integers from $0$ to any $N$ in constant memory.
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Linear Congruential Generators

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If you pick $a$, $b$, and $N$ correctly, then the sequence:

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$K_i = (a\cdot K_{iāˆ’1}+b)\;mod\;N$

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will produce $N$ distinct, pseudorandom integers $K_i \in [0, N)$

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Cyclic Sampling

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To marginalize $p(\{X_i\})$...

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  1. Init an LCG with a cycle of $N = \prod_i |dom(X_i)|$
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  3. Use the LCG to sample $\{x_i\} \in \{X_i\}$
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  5. Incorporate $p(x_i = X_i)$ into the OLA estimate
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  7. Repeat from 2 until done
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Cyclic Sampling

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Advantages
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Progressively better estimates over time.
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Converges in bounded time.
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Disadvantages
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Exponential time in the number of variables.
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Accuracy

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Sampling with Replacement
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Chernoff Bounds, Hoeffding Bounds give an $\epsilon-\delta$ guarantee on the sum/avg of a sample with replacement.
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Without Replacement?
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Serfling et. al. have a variant of Hoeffding Bounds for sampling without replacement.
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Better OLA: Leaky Joins

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Make Cyclic Sampling into a composable operator

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$G$#$\sum p_{\psi_2}$
130.348
240.288
340.350
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$D$$G$#$\sum p_{\psi_1}$
0110.126
1120.222
0220.238
1220.050
0320.322
1320.028
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$G$#$\sum p_{\psi_2}$
130.348
240.288
340.350
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$D$$G$#$\sum p_{\psi_1}$
0120.140
1120.222
0220.238
1220.050
0320.322
1320.028
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$G$#$\sum p_{\psi_2}$
140.362
240.288
340.350
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$D$$G$#$\sum p_{\psi_1}$
0120.140
1120.222
0220.238
1220.050
0320.322
1320.028
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Leaky Joins

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  1. Build a normal join/aggregate graph as in variable elimination: One Cyclic Sampler for each Join+Aggregate.
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  3. Keep advancing Cyclic Samplers in parallel, resetting their output after every cycle so samples "leak" through.
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  5. When the sampler completes one full cycle with a complete input, mark it complete and stop sampling it.
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  7. Continue until a desired accuracy is reached or all tables marked complete.
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There's a bit of extra math to compute $\epsilon-\delta$ bounds by adapting Serfling's results. It's in the paper.

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Experiments

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Microbenchmarks
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Fix time, vary domain size, measure accuracy
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Fix domain size, vary time, measure accuracy
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Vary domain size, measure time to completion
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Macrobenchmarks
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4 graphs from the bnlearn Repository
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Microbenchmarks

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Student: A common benchmark graph.

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Accuracy vs Domain

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VE is binary: It completes, or it doesn't.

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Accuracy vs Time

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CS gets early results faster, but is overtaken by LJ.

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Domain vs Time to 100%

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LJ is only 3-5x slower than VE.

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"Child"

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LJ converges to an exact result before Gibbs gets an approx.

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"Insurance"

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On some graphs Gibbs is better, but only marginally.

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More graphs in the paper.

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Leaky Joins

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  • Classical OLA isn't appropriate for GMs.
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  • Idea 1: LCGs can sample without replacement.
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  • Idea 2: "Leak" samples through a normal join graph.
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  • Compared to both Variable Elim. and Gibbs Sampling, Leaky Joins are often better and never drastically worse.
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