Secretary Problems: The Power of a Single Sample

In this paper, we investigate two variants of the secretary problem. In these variants, we are presented with a sequence of numbers $X_i$ that come from distributions $\mathcal{D}_i$, and that arrive in either random or adversarial order. We do not know what the distributions are, but we have access to a single sample $Y_i$ from each distribution $\mathcal{D}_i$. After observing each number, we have to make an irrevocable decision about whether we would like to accept it or not with the goal of maximizing the probability of selecting the largest number. The random order version of this problem was first studied by Correa et al. [SODA 2020] who managed to construct an algorithm that achieves a probability of $0.4529$. In this paper, we improve this probability to $0.5009$, almost matching an upper bound of $\simeq 0.5024$ which we show follows from earlier work. We also show that there is an algorithm which achieves the probability of $\simeq 0.5024$ asymptotically if no particular distribution is especially likely to yield the largest number. For the adversarial order version of the problem, we show that we can select the maximum number with a probability of $1/4$, and that this is best possible. Our work demonstrates that unlike in the case of the expected value objective studied by Rubinstein et al. [ITCS 2020], knowledge of a single sample is not enough to recover the factor of success guaranteed by full knowledge of the distribution.