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This paper considers a generalized version of the coin weighing problem with a spring scale that lies at the intersection of group testing and compressed sensing problems. Given a collection of n ≥ 2 coins of total weight d (for a known integer d), where the weight of each coin is an unknown integer in the range of {0, 1, ..., k} (for a known integer k ≥ 1), the goal is to determine the weight of each coin by weighing subsets of coins in a spring scale. The problem is to devise a weighing strategy that minimizes the average number of weighings over all possible weight configurations. For d = k = 1, an adaptive bisecting weighing strategy is known to be optimal. However, even the simplest non-trivial case of the problem, i.e., d = k = 2, is still open. For this case, we propose and analyze a simple and effective adaptive weighing strategy. Our analysis shows that the proposed strategy requires about 1.365log2n-0.5 weighings on average. As n grows unbounded, the proposed strategy, when compared to an optimal strategy within the commonly-used class of nested strategies, requires about 31.75% less number of weighings on average; and in comparison with the information-theoretic lower bound, it requires at most about 8.16% extra number of weighings on average.
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Optimal adaptive algorithms for estimating mixtures of unknown coins.
Given a mixture between two populations of coins, “positive” coins that each have unknown and potentially different—bias ≥ 1 + ∆ and “negative” coins with bias ≤ 2 − ∆, we consider the task of estimating the fraction ρ of positive coins to within additive error E. We achieve an upper and lower bound of Θ( ρ log 1 ) samples for a 1 −δ probability of success, where crucially, our lower bound applies to all fully-adaptive algorithms. Thus, our sample complexity bounds have tight dependence for every relevant problem parameter. A crucial component of our lower bound proof is a decomposition lemma (Lemma 5.2) showing how to assemble partially-adaptive bounds into a fully-adaptive bound, which may be of independent interest: though we invoke it for the special case of Bernoulli random variables (coins), it applies to general distributions. We present sim- ulation results to demonstrate the practical efficacy of our approach for realistic problem parameters for crowdsourcing applications, focusing on the “rare events” regime where ρ is small. The fine-grained adaptive flavor of both our algo- rithm and lower bound contrasts with much previous workin distributional testing and learning.
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- Award ID(s):
- 1900460
- PAR ID:
- 10273447
- Date Published:
- Journal Name:
- Proceedings of the 2021 ACM- SIAM Symposium on Discrete Algorithms (SODA), pages 414–433.
- Volume:
- SIAM 2021
- Issue:
- 2021
- Page Range / eLocation ID:
- 414-433
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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