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Identifying Approximate Minimizers under Stochastic Uncertainty

Hessa Al-Thani, Viswanath Nagarajan

TL;DR

This work tackles the stochastic minimum query problem where one must identify a $\delta$-approximate minimum (or maximizer) among $n$ independent, interval-bounded random variables with query costs. It introduces non-adaptive policies that interleave two natural greedy criteria, achieving constant-factor approximations: 4-approximation for unit costs in both SMQ and SMQI, 5.83-approximation for SMQ with non-uniform costs, and 7.47-approximation for SMQI with non-uniform costs. The analysis hinges on stopping-probability comparisons and stochastic dominance, and extends from unit-cost to general-cost settings using power-of-two iterations and a knapsack subroutine. A key contribution is showing that non-adaptive policies can closely approximate adaptive-optimal performance, providing explicit adaptivity-gap bounds and enabling practical, implementable procedures for stochastic exploration under uncertainty. The results advance constant-factor guarantees in stochastic probing and exploration problems, with concrete implications for cost-aware design choices in uncertain environments.

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 value that is $δ$-approximately minimum (or maximum) 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.

Identifying Approximate Minimizers under Stochastic Uncertainty

TL;DR

This work tackles the stochastic minimum query problem where one must identify a -approximate minimum (or maximizer) among independent, interval-bounded random variables with query costs. It introduces non-adaptive policies that interleave two natural greedy criteria, achieving constant-factor approximations: 4-approximation for unit costs in both SMQ and SMQI, 5.83-approximation for SMQ with non-uniform costs, and 7.47-approximation for SMQI with non-uniform costs. The analysis hinges on stopping-probability comparisons and stochastic dominance, and extends from unit-cost to general-cost settings using power-of-two iterations and a knapsack subroutine. A key contribution is showing that non-adaptive policies can closely approximate adaptive-optimal performance, providing explicit adaptivity-gap bounds and enabling practical, implementable procedures for stochastic exploration under uncertainty. The results advance constant-factor guarantees in stochastic probing and exploration problems, with concrete implications for cost-aware design choices in uncertain environments.

Abstract

We study a fundamental stochastic selection problem involving independent random variables, each of which can be queried at some cost. Given a tolerance level , the goal is to find a value that is -approximately minimum (or maximum) 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 -minimum value or a -minimizer. When all query costs are uniform, we provide a -approximation algorithm for both variants. When query costs are non-uniform, we provide a -approximation algorithm for the -minimum value and a -approximation for the -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.

Paper Structure

This paper contains 32 sections, 19 theorems, 60 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Adaptive and Non-adaptive policies
  4. Results
  5. Related Work
  6. Preliminaries
  7. Stopping rule for ${\sf SMQ}$.
  8. Adaptivity gap.
  9. Fixed threshold problem.
  10. Algorithm for Unit Costs
  11. Proof of Key Lemma
  12. Finding the minimum interval
  13. Remark:
  14. Stopping criteria for
  15. Handling the old stopping criterion.
  16. ...and 17 more sections

Key Result

Proposition 1.1

A policy for ${\sf SMQ}$ can stop if and only if criterion eq:smq-stop-rule holds.

Figures (5)

  • Figure 1: Adaptivity gap instance for ${\sf SMQ}$.
  • Figure 2: Optimal adaptive policy
  • Figure 5: Illustration of new stopping criterion.
  • Figure 6: Illustration of Definition \ref{['def:S']}.
  • Figure 7: Adaptive policy ${\cal A}$ for the fixed threshold problem.

Theorems & Definitions (37)

  • Proposition 1.1
  • Proposition 1.2
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 27 more