thank you, will look into that. I intuitively expect that in the setting where compute is precisely 0 cost, you can always just convert multiplicity to negative-length by building an iterate/sort/index loop around the bit segment where the multiplicity lies, and this just costs you the length of the iterate/sort/index loop (a constant which depends on your language). I also intuitively expect this to break in the infinite bitstring setting because you can have multiplicity that isn't contained in a finite substring?
I was not able on a quick skim of the pdf to identify which passage you were referring to. If possible can you point me to an example Temperature 0 in the textbook?
The post's claim that validation-only approaches are fundamentally better than training-with-validation oversimplifies a complex reality. Both approaches modify the distribution of models - neither preserves some "pure" average case. Our base training objective may already have some correlation with our validation signal, and there's nothing special about maintaining this arbitrary starting point. Sometimes we should increase correlation between training and validation, sometimes decrease it, depending on the specific relationship between our objective and validator. What matters is understanding how correlation affects both P(aligned) and P(pass|misaligned), weighing the tradeoffs, and optimizing within our practical retraining budget (because often, increasing P(aligned|pass) will also decrease P(pass)).