Identifying Approximate Minimizers under Stochastic Uncertainty

2025-04-23

AI Summary

This paper studies the problem of identifying a δ-approximate minimum (δ-minimum) or δ-minimizer with minimal expected query cost in a sequence of random variables, under both adaptive and non-adaptive querying strategies. Addressing the more realistic setting of non-uniform query costs, we design the first polynomial-time algorithms achieving approximation ratios of 5.83 for δ-minimum and 7.47 for δ-minimizer; under uniform costs, our algorithms attain the optimal 4-approximation. Our approach integrates stochastic dominance analysis, stopping probability modeling, and constructive non-adaptive strategy design. Crucially, we rigorously quantify the adaptivity gap—proving that the benefit of adaptivity is bounded by a deterministic constant. These results establish the first systematic theoretical framework for approximate selection in stochastic optimization, accompanied by tight approximation guarantees.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $delta$, the goal is to find a value that is $delta$-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 $delta$-minimum value or a $delta$-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 $delta$-minimum value and a $7.47$-approximation for the $delta$-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.

Problem

• Find δ-approximately minimum value with minimal cost
• Develop adaptive query algorithms for stochastic selection
• Analyze adaptivity gaps for non-uniform query costs

Innovation

• Adaptive query sequence for stochastic optimization
• 4-approximation algorithm for uniform costs
• Non-uniform cost approximation algorithms