Dynamically choosing the pairs to compare in an experiment to limit the number of comparisons about choix HOT 10 CLOSED

This discussion is interesting for us (tensorflow/minigo). We're using (2) to do dynamic pairing of different NN models which are very expensive to evaluate (~1-3minutes) per eval.

We do a variation on active learning where we select a couple of points that maximize expected information gain and evaluate them. We stop not after k but after std_err is below some threshold.

I've been evaluating the goodness of ranking via the inverse of the hessian (justification was this post I think)

        hessian = choix.opt.PairwiseFcts(pairs, penalty=.1).hessian(ilsr_param)
        std_err = np.sqrt(np.diagonal(np.linalg.inv(hessian)))

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@lucasmaystre random.sample(range(M, 2) is clearly better, that was a silly oversight on my part.

which is in some cases (marginally) more accurate. But also more expensive, since you need multiple matrix inversions.
I misread the "marginally" as applying to both statements (accuracy, and cost).

I converted the script to a Colab (colab is a Google hosted + more magical version of Jupyter notebooks) and printed some plots about shift in ratings and std_err and speed.

TL;DR ep_pairwise is very similar but 20-30x slower. which is to slow for my use case :/

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@sethtroisi very nice. What you are doing can also be thought of as computing the Laplace approximation of the posterior distribution.

An alternative would be:

mean, cov = choix.ep_pairwise(n_items, pairs, 0.1, model="logit")
std_err = np.sqrt(np.diagonal(cov))

This computes an approximate posterior using the EP algorithm, which is in some cases (marginally) more accurate. But also more expensive, since you need multiple matrix inversions.

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Hi @wegel, that's a great question. Supposing that you have a budget of k comparisons, there are two ways to go about it:

1. selecting all k comparisons before observing the outcomes.
2. iteratively & adaptively select pairs of items to compare based on the outcomes of previous comparisons

For (1), there is some theory that suggests that you cannot do much better that selecting k pairs uniformly at random among all n² pairs.

For (2), a.k.a. the active learning setting, there is a number of approaches, but it is still very much an active area of research. Some alternatives:

• In my opinion, the best approach is to greedily maximize the expected information gain, i.e., to follow the approach of this paper. But the cost of this is O(n³), so not sure if it'll work in your case. I have some code for this if you want to try - let me know and I'll add you to the (private) repo.
• Another approach is to use a search algorithm (such as Mergesort) as a "heuristic" to select pairs. This idea is explained in this paper (which is part of my PhD thesis).
• This recent paper seems to provide another approach.

Hope this helps!

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Interesting. If I chose (1), how would I know (potentially using choix) that the randomly selected pairs were sufficient to calculate with good certainty the rank of all items? Would I just call choix.ilsr_pairwise and check if it raises a RuntimeError: Did not converge after 100 iterations, and if did raise the error, maybe select a few other random pairs to compare? How can I determine that the ranking truthfulness has improved at each iteration in that case?

Intuitively, using something similar to quicksort for this problem seems like a good fit, so I'll read your paper with attention (and try to understand how I would actually implement/program it). Thanks!

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Regarding your first point, I don't think there is any ideal way to do this. In particular, convergence is not necessarily the right thing to look at. What I would look at is as follows:

• stability: if I train the model on part of the data vs. all of the data, does the ranking change by much? (you can use choix.{footrule,kendall}_dist or look at the norm of the difference between parameter vectors)
• predictive accuracy: splitting the data the data into two sets A, B, how well does a model trained on A predict comparison outcomes in B?

In either case, it is not clear when to stop adding pairs - at some point you will probably just need to decide that the results are "good enough".

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Thanks for the links. I'll try it out. We are slightly sensitive to speed, we have 1.5M pairs over ~10K models takes several minutes to converge and invert.

A related question I've been curious about.
What does penalty represent? Is there some reasonable way to choice it?

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@sethtroisi penalty is simply the L2-regularization hyperparameter. In the inference functions exposed in the top-level package, it is called alpha.

In other words: if you minimize choix.opt.PairwiseFcts(pairs, penalty=x).objective (which you would do when calling choix.opt_pairwise(n_items, pairs, alpha=x)) then you are computing the maximum a-posteriori estimate of the parameters, under a spherical Gaussian prior with variance 1/x.

If you can make a guess about the range of the parameters, you can use that to set penalty. Otherwise, you can estimate the best value using cross-validation or similar. Informally, 0.1 seems like a decent default value :-)

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@lucasmaystre My math background is more in discrete math so I'm Googling everything you say to trying to understand. Thanks again for following up.

ep_pairwise seems to always throw an error I wrote up this simple test program to demonstrate

I'm using this to power the ratings page for Minigo, on this data set (~150,000 games) it ran for >10 minutes before I killed it. I tried with a smaller data set (~10000 games but it still

my params have a range of ~25
with penalty 0.1 my std_err go from 0.09 to 1.80
with penalty 0.05 my std_err go from 0.11 to 2.19
with penalty 0.2 my std_err go from 0.08 to 1.40

I guess I'll spend sometime seeing what minimizes error on a holdout game set.

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@sethtroisi regarding your test program: some of the pairs you generate on line 16 have twice the same player. This throws ep_pairwise off. If you replace line 16 by:

players = random.sample(range(M), 2)

I think it should work.

In general I would expect ep_pairwise to be significantly slower than what you used until now (by a factor 10x-100x). There are ways to speed this up, but I'm not sure it's worth it - simply using the inverse hessian at the maximum-likelihood estimate of the parameters, like you did so far, should be good enough in practice.

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