TY - GEN

T1 - Approximate modularity revisited

AU - Feige, Uriel

AU - Feldman, Michal

AU - Talgam-Cohen, Inbal

N1 - Publisher Copyright:
© 2017 Copyright held by the owner/author(s).

PY - 2017/6/19

Y1 - 2017/6/19

N2 - Set functions with convenient properties (such as submodularity) appear in application areas of current interest, such as algorithmic game theory, and allow for improved optimization algorithms. It is natural to ask (e.g., in the context of data driven optimization) how robust such properties are, and whether small deviations from them can be tolerated. We consider two such questions in the important special case of linear set functions. One question that we address is whether any set function that approximately satisfies the modularity equation (linear functions satisfy the modularity equation exactly) is close to a linear function. The answer to this is positive (in a precise formal sense) as shown by Kalton and Roberts [1983] (and further improved by Bondarenko, Prymak, and Radchenko [2013]). We revisit their proof idea that is based on expander graphs, and provide significantly stronger upper bounds by combining it with new techniques. Furthermore, we provide improved lower bounds for this problem. Another question that we address is that of how to learn a linear function h that is close to an approximately linear function f, while querying the value of f on only a small number of sets. We present a deterministic algorithm that makes only linearly many (in the number of items) nonadaptive queries, by this improving over a previous algorithm of Chierichetti, Das, Dasgupta and Kumar [2015] that is randomized and makes more than a quadratic number of queries. Our learning algorithm is based on a Hadamard transform.

AB - Set functions with convenient properties (such as submodularity) appear in application areas of current interest, such as algorithmic game theory, and allow for improved optimization algorithms. It is natural to ask (e.g., in the context of data driven optimization) how robust such properties are, and whether small deviations from them can be tolerated. We consider two such questions in the important special case of linear set functions. One question that we address is whether any set function that approximately satisfies the modularity equation (linear functions satisfy the modularity equation exactly) is close to a linear function. The answer to this is positive (in a precise formal sense) as shown by Kalton and Roberts [1983] (and further improved by Bondarenko, Prymak, and Radchenko [2013]). We revisit their proof idea that is based on expander graphs, and provide significantly stronger upper bounds by combining it with new techniques. Furthermore, we provide improved lower bounds for this problem. Another question that we address is that of how to learn a linear function h that is close to an approximately linear function f, while querying the value of f on only a small number of sets. We present a deterministic algorithm that makes only linearly many (in the number of items) nonadaptive queries, by this improving over a previous algorithm of Chierichetti, Das, Dasgupta and Kumar [2015] that is randomized and makes more than a quadratic number of queries. Our learning algorithm is based on a Hadamard transform.

KW - Learning

KW - Linear functions

KW - Set functions

UR - http://www.scopus.com/inward/record.url?scp=85024382816&partnerID=8YFLogxK

U2 - 10.1145/3055399.3055476

DO - 10.1145/3055399.3055476

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AN - SCOPUS:85024382816

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1028

EP - 1041

BT - STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

A2 - McKenzie, Pierre

A2 - King, Valerie

A2 - Hatami, Hamed

PB - Association for Computing Machinery

T2 - 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017

Y2 - 19 June 2017 through 23 June 2017

ER -