The Multisecretary problem with many types
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We study the multisecretary problem with capacity to hire up to B out of T candidates, and values drawn i.i.d. from a distribution F on [0,1]. We investigate achievable regret performance, where the latter is defined as the difference between the performance of an oracle with perfect information of future types (values) and an online policy. While the case of distributions over a few discrete types is well understood, very little is known when there are many types, with the exception of the special case of a uniform distribution of types. In this work we consider a larger class of distributions which includes the few discrete types as a special case. We first establish the insufficiency of the common certainty equivalent heuristic for distributions with many types and "gaps" (intervals) of absent types; even for simple deviations from the uniform distribution, it leads to regret Θ(√(T)), as large as that of a non-adaptive algorithm. We introduce a new algorithmic principle which we call "conservativeness with respect to gaps" (CwG), and use it to design an algorithm that applies to any distribution. We establish that the proposed algorithm yields optimal regret scaling of Θ̃(T^1/2 - 1/(2(β + 1))) for a broad class of distributions with gaps, where β quantifies the mass accumulation of types around gaps. We recover constant regret scaling for the special case of a bounded number of types (β=0 in this case). In most practical network revenue management problems, the number of types is large and the current certainty equivalent heuristics scale poorly with the number of types. The new algorithmic principle called Conservatism w.r.t Gaps (CwG) that we developed, can pave the way for progress on handling many types for the broader class of network revenue management problems like order fulfillment and online matching.
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