research-article

An improved cutting plane method for convex optimization, convex-concave games, and its applications

Authors:
Haotian Jiang

University of Washington, USA

University of Washington, USA
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,
Yin Tat Lee

University of Washington, USA / Microsoft Research, USA

University of Washington, USA / Microsoft Research, USA
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,
Zhao Song

Institute for Advanced Study at Princeton, USA / Princeton University, USA

Institute for Advanced Study at Princeton, USA / Princeton University, USA
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,
Sam Chiu-wai Wong

Microsoft Research, USA

Microsoft Research, USA
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STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of ComputingJune 2020Pages 944–953https://doi.org/10.1145/3357713.3384284

Published:22 June 2020Publication History

STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

Pages 944–953

ABSTRACT

Given a separation oracle for a convex set K ⊂ ℝⁿ that is contained in a box of radius R, the goal is to either compute a point in K or prove that K does not contain a ball of radius є. We propose a new cutting plane algorithm that uses an optimal O(n log(κ)) evaluations of the oracle and an additional O(n ²) time per evaluation, where κ = nR/є.

This improves upon Vaidya’s O( SO · n log(κ) + n ^ω+1 log(κ)) time algorithm [Vaidya, FOCS 1989a] in terms of polynomial dependence on n, where ω < 2.373 is the exponent of matrix multiplication and SO is the time for oracle evaluation. This improves upon Lee-Sidford-Wong’s O( SO · n log(κ) + n ³ log^O(1) (κ)) time algorithm [Lee, Sidford and Wong, FOCS 2015] in terms of dependence on κ.

For many important applications in economics, κ = Ω(exp(n)) and this leads to a significant difference between log(κ) and (log(κ)). We also provide evidence that the n ² time per evaluation cannot be improved and thus our running time is optimal.

A bottleneck of previous cutting plane methods is to compute leverage scores, a measure of the relative importance of past constraints. Our result is achieved by a novel multi-layered data structure for leverage score maintenance, which is a sophisticated combination of diverse techniques such as random projection, batched low-rank update, inverse maintenance, polynomial interpolation, and fast rectangular matrix multiplication. Interestingly, our method requires a combination of different fast rectangular matrix multiplication algorithms.

Our algorithm not only works for the classical convex optimization setting, but also generalizes to convex-concave games. We apply our algorithm to improve the runtimes of many interesting problems, e.g., Linear Arrow-Debreu Markets, Fisher Markets, and Walrasian equilibrium.

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Index Terms

An improved cutting plane method for convex optimization, convex-concave games, and its applications
1. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Convex optimization

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Published in
STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
June 2020
1429 pages
ISBN:9781450369794
DOI:10.1145/3357713
General Chairs:
Konstantin Makarychev
Northwestern University, USA
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Yury Makarychev
Toyota Technological Institute at Chicago, USA
,
Madhur Tulsiani
Toyota Technological Institute at Chicago, USA
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Gautam Kamath
University of Waterloo, Canada
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Program Chair:
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Toyota Technological Institute at Chicago, USA
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convex optimization
convex-concave games
cutting plane method
fast rectangular matrix multiplication
market equilibrium
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An improved cutting plane method for convex optimization, convex-concave games, and its applications

STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

ABSTRACT

References

Cited By

Index Terms

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