[Paper Review] Robust Algorithms for the Secretary Problem

Domagoj Bradač, Anupam Gupta+2 more|arXiv (Cornell University)|Nov 17, 2019

Optimization and Search Problems36 references4 citations

TL;DR

This paper introduces the Byzantine Secretary problem, a robust variant of the classical secretary problem where adversarial 'red' elements corrupt the random arrival order, while 'green' elements still arrive uniformly at random. The authors present algorithms that achieve near-optimal performance by competing against a benchmark that subtracts the largest green element's value, ensuring resilience against adversarial outliers in both single-item and knapsack settings.

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 $1/e$-secretary algorithm fails with even a single adversarial arrival. 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 give algorithms which get value comparable to the value of the optimal green set minus the largest green item. Specifically, we give an algorithm to pick $K$ elements that gets within $(1-\varepsilon)$ factor of the above benchmark, as long as $K \geq \mathrm{poly}(\varepsilon^{-1} \log n)$. We extend this to the knapsack secretary problem, for large knapsack size $K$. For the single-item case, an analogous benchmark is the value of the second-largest green item. For value-maximization, we give a $\mathrm{poly} \log^* n$-competitive algorithm, using a multi-layered bucketing scheme that adaptively refines our estimates of second-max over time. For probability-maximization, we show the existence of a good randomized algorithm, using the minimax principle.

Motivation & Objective

• To address the fragility of classical secretary algorithms under adversarial perturbations in the random-order model.
• To design algorithms that remain effective even when a constant fraction of arrivals are adversarially placed (red items), while the rest (green items) arrive uniformly at random.
• To establish a robust performance benchmark that subtracts the largest green item's value, reflecting regret-minimization and fairness in sequential decision-making.
• To extend robustness to the knapsack secretary problem and both value-maximization and probability-maximization objectives.
• To demonstrate the existence of competitive algorithms using minimax principles and discretization techniques.

Proposed method

• Propose the Byzantine Secretary model with two types of elements: green (randomly arriving) and red (adversarially arriving), both with values chosen by an adversary.
• Introduce a performance benchmark that compares the algorithm’s outcome to the optimal green set minus its largest element, enabling robustness to outliers.
• Use a multi-layered bucketing scheme to estimate the second-highest green value in the single-item case, achieving poly(log* n)-competitiveness.
• Apply discretization of the continuous time interval [0,1] into N = n³ points to make the set of possible histories and algorithms finite, enabling application of Von Neumann’s minimax theorem.
• Leverage the minimax principle to prove the existence of a randomized algorithm for probability-maximization, even without explicit construction.
• Extend the framework to the knapsack secretary problem for large knapsack size K ≥ poly(ε⁻¹ log n), achieving (1−ε)-approximation to the benchmark.

Experimental results

Research questions

• RQ1 Can we design secretary algorithms that remain competitive when a constant fraction of arrivals are adversarially placed, even if they corrupt the random order assumption?
• RQ2 What is a meaningful robust benchmark for performance in the presence of adversarial outliers, and can we achieve near-optimality relative to it?
• RQ3 Can we achieve (1−ε)-approximation to the optimal green set minus its largest element in the single-item and knapsack secretary problems?
• RQ4 Is there a poly(log* n)-competitive algorithm for value-maximization in the single-item Byzantine secretary problem?
• RQ5 Can we prove the existence of a good randomized algorithm for probability-maximization using game-theoretic principles?

Key findings

• The paper establishes that selecting the highest-value red set or the single largest green element is impossible to achieve even with a small fraction of red items.
• For the single-item case, a poly(log* n)-competitive algorithm is developed for value-maximization using a multi-layered bucketing scheme to estimate the second-largest green value.
• For knapsack secretary problems with large K ≥ poly(ε⁻¹ log n), the paper achieves a (1−ε)-approximation to the benchmark of the optimal green set minus its largest element.
• The existence of a good randomized algorithm for probability-maximization is proven via the minimax principle, despite the lack of an explicit construction.
• By discretizing the time interval [0,1] into n³ points, the authors reduce the infinite space of algorithms and inputs to a finite one, enabling application of Von Neumann’s minimax theorem.