Learning the Hidden Set Locally
Primary Area: learning theory
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Keywords: hidden set, group testing, local testing, non-adaptive queries, deterministic algorithms, lower bounds, clusters
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TL;DR: Non-adaptive Group Testing from new angles of locality and (cluster) avoidance - efficient constructions and lower bounds.
Abstract: Learning elements of the hidden set(s), also known as group testing (GT), is a well-established area in which one party tries to discover elements hidden by the other party by asking queries and analyzing feedback. The feedback is a function of the intersection of the query with the hidden set -- in our case, it is a classical double-threshold function, which returns $i$ if the intersection is a singleton $i\in [n]$ and "null" otherwise (i.e., when the intersection is empty or of size at least $2$). In this work, we introduce a local framework to this problem: each hidden element is an "autonomous" element and can analyze feedback itself, but only for the queries which this element is a part of. The goal is to design a deterministic non-adaptive sequence of queries that allows each non-hidden element to learn about all other hidden agents. We show that, surprisingly, this task requires substantially more queries than the classic group testing -- by proving a super-qubic (in terms of the number of hidden elements) lower bound and constructing a specific sequence of slightly longer length. We also extend the results to the model, where agents belong to various clusters and selection must be done in queries avoiding elements from ``interfering'' clusters. Our algorithms could be generalized to other feedback functions, to adversarial/stochastic fault-prone scenarios and applied to codes.
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Submission Number: 7802
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