Hey - my name is Mike Johanson, I'm one of the PhD students at the University of Alberta that developed Hyperborean and Polaris. I was emailed and asked to drop into the thread to say whether heads-up limit Texas hold'em has been solved yet. I haven't read the rest of the thread, so my apologies if I cover something that's already been discussed.
Heads-up limit isn't solved yet. A couple of years ago, I posted in the "Math is apparently not important" thread
(link) thread to discuss a technique we had for measuring how close a strategy was to a Nash equilibrium by computing how much it can lose against a perfect adversary. A Nash equilibrium would lose $0 / game, and the lower your exploitability is, the closer you are to Nash.
In that 2011 thread, I mentioned that the strategies we had for the 2008 Man-vs-Machine match were beatable for 235 milli-big-blinds / game, or 11.75 BB/100 (just divide by 20 to convert). The best strategies we had in 2011 were quite a bit better, being exploitable for 104.41 mbb/g, or about 5 BB/100.
We've made a lot of progress since then. Our most recent game solving algorithm, called CFR-BR
(link) lets us use abstraction techniques but get as close as possible to a Nash equilibrium within an abstraction. In that paper we had strategies as low as 41.199 mbb/g (2 BB/100), and we've made more progress since then. We didn't use this strategy in the Annual Computer Poker Competition this year; our heads-up limit entry in 2012 and 2013 was the same 2011 entry I mentioned in that previous post that's beatable for 104 mbb/g. So at least for the University of Alberta - we're getting close to equilibrium, but we aren't there yet. We haven't solved the game.
Neo Poker Bot won the Annual Computer Poker Competition heads-up limit event this year. They did great - I'd be curious to know more about how their system works. In their description of their agent on the ACPC website
(link), they said that they did game tree search without approximation. This doesn't mean that they've solved the game; game tree search is a different problem than game solving, and search is way easier than solving. For example, you could use a Monte-Carlo Tree Search technique to search the game without using abstraction, and it wouldn't necessarily even take much memory, but it wouldn't converge to a Nash equilibrium (ie, solve the game) even in the limit. So their bot is clearly good, but as I understand it I don't think they're claiming to have solved the game.
I'd need to see some pretty serious proof before I'd believe anyone's claim of having solving heads-up limit. To actually brute-force your way to an exploitability of very nearly 0 in heads-up limit without using approximation techniques, the state-of-the-art CFR algorithm would need 523 terabytes of memory (probably RAM, but possibly disk) and our rough estimate is 100,000 core-years of computation. We think it's way more likely to be done first by using abstraction techniques that are good enough to get you to a tiny exploitability of under, say, 1 mbb/g. Once you're that close, the game is solved for all practical purposes; someone that could win that 1 mbb/g would have to play 100 million games to prove it with statistical confidence.