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gjm comments on The correct response to uncertainty is *not* half-speed - Less Wrong
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Randomising the length of the first step will improve on the constant factor by about 2. Similar analysis to the non-adversarial case, and with the same ETA I just added to my earlier comment.
If you make the run lengths increase exponentially instead of linearly then you get O(k) unconditionally.
I know. I was improving the constant factor.
Ah yes, OK, essentially the same randomization helps against an adversary when you're growing exponentially as when you're growing only linearly. Fair enough.