The last success problem is an optimal stopping problem that aims to maximize the probability of stopping on the last success in a sequence of independent n Bernoulli trials. In the classical setting where complete information about the distributions is available, Bruss [Bruss, 2000] provided an optimal stopping policy that ensures a winning probability of 1/e. However, assuming complete knowledge of the distributions is unrealistic in many practical applications. This paper investigates a variant of the last success problem where samples from each distribution are available instead of complete knowledge of them. When a single sample from each distribution is allowed, we provide a deterministic policy that guarantees a winning probability of 1/4. This is best possible by the upper bound provided by Nuti and Vondrák [Nuti and Vondr{á}k, 2023]. Furthermore, for any positive constant ε, we show that a constant number of samples from each distribution is sufficient to guarantee a winning probability of 1/e-ε.
@InProceedings{yoshinaga_et_al:LIPIcs.ESA.2024.105, author = {Yoshinaga, Toru and Kawase, Yasushi}, title = {{The Last Success Problem with Samples}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {105:1--105:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.105}, URN = {urn:nbn:de:0030-drops-211762}, doi = {10.4230/LIPIcs.ESA.2024.105}, annote = {Keywords: The Last Success Problem, Secretary Problem, Sample Information Model, Optimal Stopping, Online Algorithms} }
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