259 lines
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259 lines
9.1 KiB
HTML
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<title>Leaky Joins</title>
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<meta name="description" content="Convergent Interactive Inference with Leaky Joins">
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<meta name="author" content="Oliver Kennedy">
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<!-- Any Talk-Specific Header Content Goes Here -->
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<center>
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<a href="http://www.buffalo.edu" target="_blank">
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<img src="../graphics/logos/ub-1line-ro-white.png" height="20"/>
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</a>
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<!-- Any Talk-Specific Footer Content Goes Here -->
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<div style="float: left; margin-top: 15px; ">
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Exploring <u><b>O</b></u>nline <u><b>D</b></u>ata <u><b>In</b></u>teractions
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</div>
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<a href="http://odin.cse.buffalo.edu" target="_blank">
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<img src="../graphics/logos/odin-1line-white.png" height="40" style="float: right;"/>
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</a>
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</div>
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<div class="slides">
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<section>
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<section>
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<h1>Leaky Joins</h1>
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<h3><u>Ying Yang</u>, Oliver Kennedy</h3>
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</section>
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<section>
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<h1>Leaky Joins</h1>
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<h3>Ying Yang, <u>Oliver Kennedy</u></h3>
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</section>
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<section>
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<img src="graphics/yingyang.jpg" style="float: right; margin-left: 20px;" />
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<h3>Disclaimer</h3>
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<p>Ying could not be here today. If you like her ideas, get in touch with her. </p>
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<p class="fragment">(she's on the job market)</p>
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<p class="fragment">(If you don't, blame my presentation)</p>
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</section>
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</section>
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<section>
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<section>
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<h2>Online Aggregation (OLA)</h2>
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<p style="margin-top: 60px;">$Avg(3,6,10,9,1,3,9,7,9,4,7,9,2,1,2,4,10,8,9,7) = 6$</p>
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<p class="fragment">$Avg(3,6,10,9,1) = 5.8$ <span class="fragment">$\approx 6$</span></p>
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<p class="fragment">$Sum\left(\frac{k}{N} Samples\right) \cdot \frac{N}{k} \approx Sum(*)$</p>
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<p class="fragment" style="font-weight: bold; margin-top: 60px;">Sampling lets you approximate aggregate values with orders of magnitude less data.</p>
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</section>
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<section>
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<h2>Typical OLA Challenges</h2>
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<dl>
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<dt>Birthday Paradox</dt>
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<dd>$Sample(R) \bowtie Sample(S)$ is likely to be empty.</dd>
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<dt>Stratified Sampling</dt>
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<dd>It doesn't matter how important they are to the aggregate, rare samples are still rare.</dd>
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<dt>Replacement</dt>
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<dd> Does the sampling algorithm converge exactly or asymptotically?</dd>
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</dl>
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</section>
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<section>
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<h2>Replacement</h2>
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<dl>
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<dt>Sampling Without Replacement</dt>
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<dd>... eventually converges to a precise answer.</dd>
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<dt>Sampling With Replacement</dt>
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<dd>... doesn't need to track what's been sampled.</dd>
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<dd>... produces a better behaved estimate distribution.</dd>
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</dl>
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</section>
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</section>
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<section>
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<section>
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<img src="graphics/mimir_logo_final.png" />
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<p><a href="http://mimirdb.info">http://mimirdb.info</a></p>
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<p style="font-size: smaller" class="fragment">(not immediately relevant to the talk, but you should check it out)</p>
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</section>
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<section>
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<h2>Roughly 1-2 years ago...</h2>
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<p><b>Ying</b>: To implement {cool feature in Mimir}, we'll need to be able to perform <u>inference</u> on <u>Graphical Models</u>, but we will <u>not know how complex they are</u>.</p>
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</section>
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<section>
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<h2>Graphical Models</h2>
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<p>Joint probability distributions are expensive to store<br/>
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$$p(D, I, G, S, J)$$</p>
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<p>Bayes rule lets us break apart the distribution<br/>
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$$= p(D, I, G, S) \cdot p(J | D, I, G, S)$$</p>
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<p>And conditional independence lets us further simplify<br/>
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$$= p(D, I, G, S) \cdot p(J | G, S)$$</p>
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<p class="fragment">This is basis for a type of graphical model called a "Bayes Net"</p>
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</section>
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<section>
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<h2>Bayesean Networks</h2>
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<svg width="500" height="400">
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<image xlink:href="graphics/studentBN.svg" x="-125" y="-90"
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height="650" width="650" />
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class="fragment" data-fragment-index="1"
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/>
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</svg>
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<p>$p(D=1, I=0, S=0, G=2, J=1)$<br/>
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<span class="fragment" data-fragment-index="1">$=\;0.5$</span>
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<span class="fragment" data-fragment-index="2">$\cdot\;0.7$</span>
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<span class="fragment" data-fragment-index="3">$\cdot\;0.95$</span>
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<span class="fragment" data-fragment-index="4">$\cdot\;0.25$</span>
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<span class="fragment" data-fragment-index="5">$\cdot\;0.8$</span>
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</p>
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</section>
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<section>
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<p>
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$p(D,I,S,G,J)$ <br/>
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$=$<br/>
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$p(D) \bowtie p(I) \bowtie p(S|I) \bowtie p(G|D,I) \bowtie p(J|G,S)$
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</p>
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</section>
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<section>
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<h2>Inference</h2>
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<p>$p(J) = \sum_{D,I,S,G}p(D,I,S,G,J)$</p>
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(aka the computing the marginal probability)
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</section>
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<section>
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<h2>Inference Algorithms</h2>
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<dl>
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<dt>Exact (e.g. Variable Elimination)</dt>
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<dd>Fast and precise, but scales poorly with graph complexity.</dd>
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<dt>Approximate (e.g. Gibbs Sampling)</dt>
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<dd>Consistent performance, but only asymptotic convergence.</dd>
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</dl>
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<p class="fragment"><b>Key Challenge</b>: For {really cool feature} we don't know whether we should use exact or approximate inference.</p>
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</section>
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<section>
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<p>Can we gracefully degrade from exact to approximate inference?</p>
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</section>
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</section>
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<section>
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<section>
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<pre><code>
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SELECT J.J, SUM(D.p * I.p * S.p * G.p * J.p) AS p
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FROM D NATURAL JOIN I NATURAL JOIN S
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NATURAL JOIN G NATURAL JOIN J
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GROUP BY J.J
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</code></pre>
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<p class="fragment">(Inference is essentially a big group-by aggregate join query)</p>
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<p class="fragment" style="font-size: smaller">(Variable elimination is Aggregate Pushdown + Join Ordering)</p>
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</section>
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<section>
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<h2>Key Idea: OLA</h2>
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<p></p>
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</section>
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</section>
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