Abstract
In classical secretary problems, a sequence of n elements arrive in a uniformly random order, and we want to choose a single item, or a set of size K. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin’s popular 1/esecretary algorithm is sensitive to even a single adversarial arrival: if the adversary gives one large bid at the beginning of the stream, the algorithm does not select any element at all.
We investigate a robust version of the secretary problem. In the Byzantine Secretary model, we have two kinds of elements: green (good) and red (rogue). The values of all elements are chosen by the adversary. The green elements arrive at times uniformly randomly drawn from [0,1]. The red elements, however, arrive at adversarially chosen times. Naturally, the algorithm does not see these colors: how well can it solve secretary problems?
We show that selecting the highest value red set, or the single largest green element is not possible with even a small fraction of red items. However, on the positive side, we show that these are the only bad cases, by giving algorithms which get value comparable to the value of the optimal green set minus the largest green item. (This benchmark reminds us of regret minimization and digital auctions, where we subtract an additive term depending on the "scale" of the problem.) Specifically, we give an algorithm to pick K elements, which gets within (1ε) factor of the above benchmark, as long as K ≥ poly(ε^{1} log n). We extend this to the knapsack secretary problem, for large knapsack size K.
For the singleitem case, an analogous benchmark is the value of the secondlargest green item. For valuemaximization, we give a poly log^* ncompetitive algorithm, using a multilayered bucketing scheme that adaptively refines our estimates of secondmax over time. For probabilitymaximization, we show the existence of a good randomized algorithm, using the minimax principle.
We hope that this work will spur further research on robust algorithms for the secretary problem, and for other problems in sequential decisionmaking, where the existing algorithms are not robust and often tend to overfit to the model.
BibTeX  Entry
@InProceedings{bradac_et_al:LIPIcs:2020:11717,
author = {Domagoj Bradac and Anupam Gupta and Sahil Singla and Goran Zuzic},
title = {{Robust Algorithms for the Secretary Problem}},
booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
pages = {32:132:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771344},
ISSN = {18688969},
year = {2020},
volume = {151},
editor = {Thomas Vidick},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/11717},
URN = {urn:nbn:de:0030drops117171},
doi = {10.4230/LIPIcs.ITCS.2020.32},
annote = {Keywords: stochastic optimization, robust optimization, secretary problem, matroid secretary, robust secretary}
}
Keywords: 

stochastic optimization, robust optimization, secretary problem, matroid secretary, robust secretary 
Collection: 

11th Innovations in Theoretical Computer Science Conference (ITCS 2020) 
Issue Date: 

2020 
Date of publication: 

06.01.2020 