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Identifying Approximate Minimizers Under Stochastic Uncertainity

Authors Hessa Al-Thani , Viswanath Nagarajan

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ACM Subject Classification
  • Theory of computation → Stochastic approximation
  • Theory of computation → Discrete optimization
Keywords
  • Approximation algorithms
  • stochastic optimization
  • selection problem

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Abstract

We study a fundamental stochastic selection problem involving n independent random variables, each of which can be queried at some cost. Given a tolerance level δ, the goal is to find a δ-approximately minimum (or maximum) value over all the random variables, at minimum expected cost. A solution to this problem is an adaptive sequence of queries, where the choice of the next query may depend on previously-observed values. Two variants arise, depending on whether the goal is to find a δ-minimum value or a δ-minimizer. When all query costs are uniform, we provide a 4-approximation algorithm for both variants. When query costs are non-uniform, we provide a 5.83-approximation algorithm for the δ-minimum value and a 7.47-approximation for the δ-minimizer. All our algorithms rely on non-adaptive policies (that perform a fixed sequence of queries), so we also upper bound the corresponding "adaptivity" gaps. Our analysis relates the stopping probabilities in the algorithm and optimal policies, where a key step is in proving and using certain stochastic dominance properties.

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Hessa Al-Thani and Viswanath Nagarajan. Identifying Approximate Minimizers Under Stochastic Uncertainity. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.8

Author Details

Hessa Al-Thani
  • Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA
Viswanath Nagarajan
  • Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA

Funding

  • Al-Thani, Hessa: This publication was made possible by the Graduate Sponsorship Research Award from the Qatar Research and Development Institute. The findings herein reflect the work, and are solely the responsibility, of the authors.
  • Nagarajan, Viswanath: Research supported in part by NSF grant CCF-2418495.

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