Does Diversity Trump Ability? An Example of the Misuse of Mathematics [pdf]

Richard Feynman has some relevant thoughts on this subject: https://www.youtube.com/watch?v=IaO69CF5mbY

This is in Notices of the AMS, which is about as mainstream (together with SIAM Review) as mathematical publications get.

Randomization improves algorithms.

This is a contentious claim. Random choices are actually very rarely the best - they are often good enough and versatile, but they are rarely the best.

For example, consider Monte Carlo integration. You get O(N^{-1/2}) convergence. If you use a deterministic set of points explicitly designed to have low discrepancy (aka "Quasi-Monte Carlo"), you can get O(N^{-1 + logarithmic stuff}) convergence.

http://www.chrisstucchio.com/blog/2014/adversarial_bandit_is...

Eliezer Yudkowsky also wrote a great critique of this issue, though I can't find it right now.

> Page’s work on diversity has been cited by NASA, the US Geological Survey, and Lawrence Berkeley Labs, among many others

Oops.

In my opinion the reason "portfolio theory" works in the cases that you mentioned is that it most perform well across a time series within a highly dynamic environment. But if one is allow to pick an optimal team for each discrete problem then it will surely outperform.

In my mind diversity will help when you have little to no a priori knowledge and then for can not predict what attributes you will need.

I believe the article you're referring to is: http://lesswrong.com/lw/vp/worse_than_random/

Choice quote: > As a general principle, on any problem for which you know that a particular unrandomized algorithm is unusually stupid - so that a randomized algorithm seems wiser - you should be able to use the same knowledge to produce a superior derandomized algorithm.

An interesting counterexample to this, an example which to me illustrates a very powerful aspect of randomness, is the monte-carlo revolution in computer Go (AI for the ancient Asian board game). For years, computer progress stagnated, as the techniques were mostly focused on encoding human knowledge into code. While most of this human knowledge is generally correct, it introduces significant bias in the way positions are evaluated. The way these rules-of-thumb interact is very hard to predict, and tree search algorithms are quite good at finding positions that are incorrectly evaluated. Because of this, the worst-case behavior of an evaluation function is much more important than it's best-case or average behavior.

Computers started making lots of progress when a new technique was used: rather than try to evaluate positions by a large set of heuristics, they were evaluated by playing random games. Go positions are quite hard to evaluate from simple rules of thumb, and this random game approach gives a much more balanced, long-term view of positions. And most importantly, there is much less bias.

Obviously, an entirely random game, with a uniform distribution over possible moves, is easy to improve upon. But computer go programmers noticed an interesting phenomenon: while certain types of knowledge incorporated into the random move distribution (to make it "more intelligent", as judged by a human) were helpful, others were not (even after taking into account the computational cost of adding the knowledge), and it wasn't always clear why. The same observation about heuristic evaluation noted above applied: having a balanced distribution of move choices, with a reasonable probabilistic lower bound of effectiveness, is more important than making an intelligent choice that is usually correct, but has unpredictable, extreme worst-case performance.

So we see that randomness does have an important property: it avoids the downside of "knowledge" that generally seems correct but can go horribly wrong in unexpected ways.

I don't know of a good writeup of this phenomenon. My understanding of it is mostly assembled from following informal discussions on the computer-go mailing list for several years. In a quick search through my gmail archives I can't find much on the subject, but here's an interesting post about related topics in the computer chess world (that incidentally doesn't talk about randomness, but illustrates well the benefits of avoiding bias): http://www.talkchess.com/forum/viewtopic.php?topic_view=thre...