I had a very interesting discussion at NIPS with Vikash Mansingka about beam search and particle filtering. Much of what is below is a result of this conversation and he deserves much credit.
In a very real sense, particle filtering can be seen (for many models) as a sort of stochastic beam search. Just to set the stage, let's talk about generic beam search:
1. Initialize a min-queue "beam" to contain a single initial hypothesis.
2. While beam contains non-complete hypothesis:
A. Take h, the lowest scoring hypothesis from the beam
B. For each h' in the neighborhood of h
I. Add h' to the beam, with some score
C. If the beam exceeds a certain capacity, drop high-scoring elements
3. Return the lowest scoring hypothesis from beam
The key variant in beam search algorithms is the score function that's used. The most naive is just path cost---how much did we have to pay to get to the current hypothesis? The "A*" way is to use path cost + admissible heuristic cost, where the admissible heuristic is an underestimate of the amount of cost remaining to complete.
There are (of course) other variants: sometimes we expand the whole beam in one step; sometimes we use multiple beams; etc. We'll come back to these shortly.
Particle filtering (using as similar terminology to the beam search as possible), on the other hand, looks something like:
1. Initialize a max-queue "beam" to contain a single initial hypothesis.
2. While beam contains non-complete hypothesis:
A. Take h, the highest scoring hypothesis from the beam
B. Sample h' from the neighborhood of h
I. Add h' to the beam, with some score
C. If the beam exceeds a certain capacity, randomly drop elements
D. If the "effective sample size" of the beam is too low, resample elements
3. Return the highest scoring hypothesis from beam
To map beam search to particle filtering, use negative log probabilities for scores. (This turns max search into min search.)
In a sense, the major thing that differs between PF and BS is that in PF everything happens randomly -- we're always sampling from a proposal distribution, rather than greedily taking the "best" hypotheses at each step.
There are, however, some minor issues: what proposal distribution is used in step 2B in PF and what does resampling do.
In general, the proposal distribution
should be the true posterior probability, but this is pretty much always intractable (otherwise we wouldn't be using approximations at all). The most common thing to do is to use a transition probability, which corresponds exactly to using a path cost in beam search. However, just as we know that using a heuristic can help in beam search, this suggest that using a proposal distribution in PF that contains something like a future cost heuristic is going to be helpful. I'm sure this has been done, but I don't think it's nearly as common as it perhaps should be, given how helpful it is in BS.
What 2D in PF does is to measure whether the samples that we have in the beam have fallen out of the true posterior---in other words, are we "off the mark." There's a straightforward way to compute this (see the wikipedia page linked above). If we are off the mark, we resample all the elements in our beam, essentially trying to get back on the right path. My experience with PF is that this is pretty essential to getting things to work, but as far as I know there's no analogue in BS land.
It may be that these two things (resampling and heuristic) are two different ways of solving the same problem and that doing both is not so helpful. But I tend to doubt it.
One other thing that I've never seen done in PF is to maintain multiple beams. Admittedly this makes the resampling step harder (maybe...I haven't worked through it), but it may be helpful. One thing, for instance, that we found in the PF for the
coalescent was that the PF tends to like to merge clusters early with no regard for how hard life is going to be later. We use resampling to try to fix this, but another way would be to use multiple beams. This is commonly done in, eg., machine translation, where a single beam doesn't do well, so we maintain one beam for each number of foreign (or English) words covered.
So what's the point? I think that by recognizing the similarity between BS and PF, we can see ways to potentially improve both. For instance, the following seem promising:
- Use resampling techniques in beam search
- Use multiple beams in particle filtering
- Use A*-style heuristics in the proposal distribution for particle filtering
An alternative suggestion would be for NLPers to pay more attention to particle filtering. It's a nice, clean probabilistic analogue to a technique that we love (or at least are forced to use).
