To clarify, "one million" is the number of programs generated by AlphaCode on the CodeContests dataset, not the CodeForces one (although the former is a superset of the latter). The results on CodeContests are reported in table 5 (page 15 of the pdf) of the DeepMind preprint [1]. The results on CodeForces are reported in table 4 of the preprint (page 14 of the pdf).
I can't find where the paper lists the number of samples drawn for the CodeForces results, but on Section 4.4 the preprint says:
Sampling from transformer models can be easily parallelized, which allowed us to scale to millions of samples per problem (...).
Note the plural. "Millions" can mean one million, or a hundred million, but, as far as I can tell, the preprint never makes it clear which it is. Note that the results on CodeForces (table 4) are averaged over 3 evaluations in each of which AlphaCode generated "millions" of samples and finally submitted 10. So it's 3 times "millions" of samples for the actual results in table 4 (CodeForces). I assume those were the top 3 of all evaluations.
The idea that the lines of code in a target program are related to the cardinality of the program space that must be searched before a target program can be found is not completely unfounded. For sure, the cardinality of the search space for programs is some function of the number of tokens that must be combined to form each program in that space. An _exponential_ function because we're talking about combinations of tokens (assuming only grammatical strings are generated it gets a bit better, but not by much). We can take lines-of-code as a rough proxy of number of tokens, and in any case it's clear that the cardinality of the set of one-line programs is less than the cardinality of the set of two-line programs, and so on.
I'm not sure if the "Update" bit above is claiming, that AlphaCode must generate the _entire_ program space of _all_ k-line programs before it can find a solution to that problem. To be honest, even I don't think that AlphaCode is _that_ bad. Having to go through the entire program space to find one program is the worst case. On the other hand, AlphaCode does perform very poorly so who knows?
Regarding talking dogs, I have never met any, but I have heard of a horse that can do arithmetic [2].
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[1] https://storage.googleapis.com/deepmind-media/AlphaCode/comp...