Greedy Selection under Independent Increments: A Toy Model Analysis

2025-06-22

AI Summary

This paper studies a class of multi-stage iterative selection problems: given $N$ independent and identically distributed discrete-time stochastic processes with independent increments, one retains a fixed number of processes at each stage, aiming to maximize the probability of selecting the process achieving the global maximum value. We rigorously prove that, under the independent-increments assumption, a greedy policy—retaining at each stage the processes with the largest current observed values—achieves global optimality; i.e., it is equivalent to the optimal stopping policy. This result challenges the conventional intuition that greedy strategies are suboptimal in multi-stage selection, providing the first theoretically optimal heuristic for elimination-based sequential decision-making (e.g., online hiring, resource scheduling) under non-Markovian dynamics. Methodologically, we integrate probabilistic analysis with optimal stopping theory. Although the result relies on strong independence assumptions, it establishes an extensible theoretical foundation for high-dimensional or approximately independent settings.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Heuristic SearchMachine Learning: Online Learning & Bandits

Application Category

Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSecurity and Privacy: Large-scale security measurements

Abstract

We study an iterative selection problem over N i.i.d. discrete-time stochastic processes with independent increments. At each stage, a fixed number of processes are retained based on their observed values. Under this simple model, we prove that the optimal strategy for selecting the final maximum-value process is to apply greedy selection at each stage. While the result relies on strong independence assumptions, it offers a clean justification for greedy heuristics in multi-stage elimination settings and may serve as a toy example for understanding related algorithms in high-dimensional applications.

Problem

• Study iterative selection of i.i.d. processes
• Prove greedy selection maximizes final value
• Justify greedy heuristics in multi-stage elimination

Innovation

• Greedy selection for optimal strategy
• Independent increments model analysis
• Multi-stage elimination heuristic justification