What is the effect of the processor type for chess infinite analysis using the current version of Stockfish?

Question

To find the best move, the dream of any chess player, chess engines are used extensively. Nowadays, the most popular chess engine is Stockfish. Many chess players, including top chess grandmasters consider Stockfish "the best chess player of the world." We, humans, are not considered the best chess players anymore.

Today is March 9, 2023. If you go the download section of Stockfish: https://stockfishchess.org/download/ You can find that the current version is Stockfish 15.1.

It is my understanding that a better proccessor can find better results in small amount of time because it can perfom calculations faster. However, my doubt is for infinite analysis without a specific time limitation. Like for example, a faster car can arrive to a destination faster, but at the end both will arrive to the same destination, unless we are talking about a space trip.

In my experience as a chess player, after you arrive with Stockfish analysis to a depth of 60, there are only insignificant differences. I consider that after a depth of 60, we arrive to something similar to the mathematical concept of asymptote or maybe more appropiate would be the concept of limit. Of course, I would like to learn if my understanding is correct. That is the reason behind this question. I am limiting this question to Stockfish to reduce the complexity of this question. What can you tell me about it?

I am asking to assume that the first analysis is performed with a Windows machine equiped with the best processor available, using the version for faster processors, version AVX2. While the second analysis is performed in a Windows machine equiped with an older processor using the version for older processors, version POPCNT. I am using only the Windows operating system, again, only to reduce the complexity of the question.

My understanding is that while the slower processor will stay at depth=60, but the faster processor will go further (maybe depth=70) but for practical purposes there will not be difference because both would have arrive to the same destination, asymptote or mathematical limit.

To support this idea, we have evidence that correspondence chess games end in a draw most of the time. To challenge this idea, we find that the chess rating of correspondence chess players is not equal. All the correspondence chess games are collected in several databases. So, we have what in statisics in called a representative sample.

Answer 1

The way the question is stated seems to conflate two concepts:

1. Does the accuracy of the engine reach an "asymptote" after depth 60 beyond which there are only "insignificant differences". There is existing research on diminishing returns with search depth but little data on current engines and such huge depths, if only because of the enormous resource requirements to generate sufficient data there. However at least empirically, in top level computer chess competition, where engines often search in excess of 40-45 ply deep, there are still clear strength differences between engines and last-ply move changes still happen, so the gain from extra depth is likely to be more than 0 Elo even for current software and search depths. I don't see any way to meaningfully answer this question further because it's not well-defined, for example it is not clear what an "insignificant difference" is.

2. Because of the reference to "processor type" and "AVX2" vs "POPCOUNT", there seems to be an implied, but not explicitly asked, question whether the type of processor has any further consequence besides more time. The answer to that is no. If one looks at an example in the commit history of Stockfish, you will see that most changes include a reference to "bench", which is the expected amount of searched nodes on a set of test positions. The development process requires that all versions on all test machines return the same answer. Because Stockfish almost exclusively uses exact (integer) arithmetic, even for its neural network, this is a workable restriction.

This being said, at least for the second question one would have to strictly add "assuming there are no bugs in the engine". Last month such a bug was found where in very rare cases some of the versions could return a different answer. Because of the above restrictions, such a bug would have to be limited to a very small amount of positions (so it is not found by normal test coverage) and have a strength impact that's 95% likely to be less than -2 Elo (because the testing process enforces this constraint before accepting any change).