I was watching the Sinquefield Cup right now, in particular the game Svidler-Caruana. I analyzed once the position they got with Stockfish 7 64bit and it suggested the move Rf3. Svidler played instead Kg3, so i played it on Arena chess, and then i went back and re analyzed the same position, but this time the engine was suggesting Kg3. Is there some kind of machine learning algorithm implemented in the engines such that, when you go deeper in the game,they improve the analysis of previous positions?
No, there is no machine learning implemented
Due to hashing, next time when you run the same analysis you might not get the same results
Some other much weaker engines do have machine learning implemented, but not Stockfish