The linear hidden subset problem for the (1+1) EA with scheduled and adaptive mutation rates

We study unbiased $(1+1)$ evolutionary algorithms on linear functions with an unknown number $n$ of bits with non-zero weight. Static algorithms achieve an optimal runtime of $O(n (\ln n)^{2+\epsilon})$, however, it remained unclear whether more dynamic parameter policies could yield better runtime guarantees. We consider two setups: one where the mutation rate follows a fixed schedule, and one where it may be adapted depending on the history of the run. For the first setup, we give a schedule that achieves a runtime of $(1\pm o(1))\beta n \ln n$, where $\beta \approx 3.552$, which is an asymptotic improvement over the runtime of the static setup. Moreover, we show that no schedule admits a better runtime guarantee and that the optimal schedule is essentially unique. For the second setup, we show that the runtime can be further improved to $(1\pm o(1)) e n \ln n$, which matches the performance of algorithms that know $n$ in advance. Finally, we study the related model of initial segment uncertainty with static position-dependent mutation rates, and derive asymptotically optimal lower bounds. This answers a question by Doerr, Doerr, and K\"otzing.