Identifying Approximate Minimizers under Stochastic Uncertainty

We study a fundamental stochastic selection problem involving $n$ independent random variables, each of which can be queried at some cost. Given a tolerance level $\delta$, the goal is to find a value that is $\delta$-approximately minimum (or maximum) over all the random variables, at minimum expected cost. A solution to this problem is an adaptive sequence of queries, where the choice of the next query may depend on previously-observed values. Two variants arise, depending on whether the goal is to find a $\delta$-minimum value or a $\delta$-minimizer. When all query costs are uniform, we provide a $4$-approximation algorithm for both variants. When query costs are non-uniform, we provide a $5.83$-approximation algorithm for the $\delta$-minimum value and a $7.47$-approximation for the $\delta$-minimizer. All our algorithms rely on non-adaptive policies (that perform a fixed sequence of queries), so we also upper bound the corresponding ''adaptivity'' gaps. Our analysis relates the stopping probabilities in the algorithm and optimal policies, where a key step is in proving and using certain stochastic dominance properties.