What's with all the "AI just proved that this or that isn't O(n (log (n))^2) but akshually O(n (log (n))^1.99999)"??

I guess it deserves respect as progress, but it just rubs me the wrong way. Like the machine did the absolute minimum to beat the previous mark.

Often times the constant (2 in this example) is a conjectured minimum, so anything below that is a noteworthy result. Think of it as breaking through some theoretical limit.

for say FFT/integer multiplication or 3SUM, we have natural algorithms that have existed a long time with a given complexity (O(n \log n) and O(n^2), respectively). Given how long these natural algorithms have been the best algorithms we have, it is natural to conjecture they are optimal. Showing an O(n(\log n)^{.99999}) algorithm exists shows that these optimality conjectures are false.

Now, there are some critiques you can have of this. Namely, it is possible that these novel algorithms have significant trade-offs that make them almost never worthwhile in practice. "Fast" matrix multiplication algorithms are typically of this form. So perhaps this all points towards a deficiency in big O notation, which can be deceptive. But, for people who care about optimizing asymptotic complexity, it is still interesting.

Many people thought it could never be less than 2. They proved that it can. What is the true value? Nobody knows, now.

to get some intuition about why this is such a big deal, look up the history of strassen's algorithm, which solved matrix multiplication in less than O(n^3). this was a truly stunning result because it seemed intuitively obvious that the output matrix had n^2 cells each of which was calculated via an independent O(n) loop over a row/column of the input matrices, so how could you do better than n^3. but once strassen proved that you could do some clever tricks and reduce the overall time to something less than O(n^3) it started an entire cottage industry of people getting better and better algorithmic bounds. the initial breakthrough was a qualitative one, independent of how much it improved things in numerical terms.

https://hideoushumpbackfreak.com/algorithms/algorithms-stras...

Tell that to the humans working on matrix multiplication who spent years of their lives getting it from n^2.3728596 to n^2.371866, only for openai to blow it away at n^2.25

You just run it again and again and again