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Why don't the developers of AlphaZero let it learn for days, or even months on end? Wouldn't its neural network become much stronger, at the cost of just occupying some computational resources for a while? It seems like this would make it indisputably the best engine in world, since right now there's still controversy on whether it's clearly better than Stockfish.
If something like this already is happening, please correct me.
Google's hardware is already very convincing. However, allowing a network running forever doesn't mean it'll play stronger chess. There is always limitation, in machine learning we say the network has reached "convergence".
Google developers must eventually stop the training, and evaluate it's performance. They would make a better model in the next iteration.
Check out the AlphaZero paper, figure 1.
As you can see from the leftmost figure, you can let AlphaZero train, but that doesn't mean it'll improve. In fact, AlphaZero peaked at a level slightly above Stockfish 8 (in other words, it'll likely lose to Stockfish 10).
I'm no expert, but I understand Leela is competitive with Stockfish 10 not because she has trained for longer, but because she's doing certain things that AlphaZero didn't do (such as "squeeze excitation" as in the latest Test 40).
