Friday, January 20, 2012

The Shonan Meeting (Part 3): Optimal Distributed Sampling

Reservoir sampling is a beautiful gems of sampling: easy to explain, and almost magic in how it works. The setting is this:
A stream of items passes by you, and your goal is to extract a sample of size $s$ that is uniform over all the elements you've seen so far.
The technique works as follows: if you're currently examining the i-th element, select it with probability 1/i, and then pick an element uniformly from the current sample to be replaced by it. I've talked about this result before, including three different ways of proving it.

But what happens if you're in a continuous distributed setting ? Now each of $k$ players is reading a stream of items, and they all talk to a coordinator who wishes to maintain a random sample of the union of streams. Let's assume for now that $s \le k$

Each player can run the above protocol and send an item to the coordinator, and the coordinator can pick a random subset from these. But this won't work ! At least, not unless each player has read in exactly the same amount of data as each other player. This is because we need to weight the sample element sent by a player with the number of elements that player has read.

It's not hard to see that each player sends roughly log n messages to the coordinator for a stream of length n. So maybe each player also annotates the element with the number of elements it has seen so far. This sort of works, but the counts could be off significantly, since a stream that doesn't send a sample might have read many more elements since the previous time it sent an update.

This can be fixed by having each player send an extra control message when its stream increases in size by a factor of 2, and that would not change the asymptotic complexity of the process, but we still don't get a truly uniform sample.

The problem with this approach is that it's trying to get around knowing $n$, the size of the stream, which is expensive to communicate in a distributed setting. So can we revisit the original reservoir method  in a 'communication friendly way' ?

Let's design a new strategy for reservoir sampling that works as follows.
Maintain a current "threshold" t. When a new item arrives, assign it a random value r between 0 and 1. If r < t, keep the new item and set t = r, else discard it.
By using the principle of deferred decisions, you can convince yourself that this does exactly the same thing as the previous strategy (because at step i, the probability of the current element being retained is its probability of r being the minimum over the set seen so far, which is 1/i). the good thing is that this approach doesn't need to know how many elements have passed so far.

This approach can be extended almost immediately to the distributed setting. Each player now runs this protocol instead of the previous one, and every time the coordinate gets an update, it sends out a new global threshold (the minimum over all thresholds sent in) to all nodes. If you want to maintain a sample of size $s$, the coordinator keeps $s$ of the $k$ elements sent in, and the overall complexity is $O(ks \log n)$.

But you can do even better.

Now, each player maintains its own threshold. The coordinator doesn't broadcast the "correct" threshold until a player sends an element whose random value is above the global threshold. This tells the coordinator that the player had the wrong threshold, and it then updates that player (and only that player)
Analyzing this approach takes a little more work, but the resulting bound is much better:
The (expected) amount of communication is $O(k \frac{\log (n/s)}{\log (k/s)})$
What's even more impressive: this is optimal !

This last algorithm and the lower bound, were presented by Srikanta Tirthapura at the Shonan meeting, based on his DISC 2011 work with David Woodruff. Key elements of this result (including a broadcast-the-threshold variant of the upper bound) also appeared in a PODS 2010 paper by Muthu, Graham Cormode, Kevin Yi and Qin Zhang. The optimal lower bound is new, and rather neat. 
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