2 Search Results for "Natarajan, Anand V."


Document
Complexity Lower Bounds for Computing the Approximately-Commuting Operator Value of Non-Local Games to High Precision

Authors: Matthew Coudron and William Slofstra

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP-complete to decide whether the classical value of a non-local game is 1 or 1- epsilon, promised that one of the two is the case. Furthermore, as long as epsilon is small enough, this result does not depend on the gap epsilon. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap epsilon decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIP^{co}(2,1,1,s) (proofs with two provers, one round, completeness probability 1, soundness probability s, and commuting-operator strategies) can increase without bound as the gap 1-s gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK-MIP^{co}_{delta}(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1-s gets arbitrarily small. While we do not know any computable time upper bound on the class MIP^{co}, a result of the first author and Vidick shows that for s = 1-1/poly(f(n)) and delta = 1/poly(f(n)), the class MIP^{co}_delta(2,1,1,s), with constant communication from the provers, is contained in TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.

Cite as

Matthew Coudron and William Slofstra. Complexity Lower Bounds for Computing the Approximately-Commuting Operator Value of Non-Local Games to High Precision. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{coudron_et_al:LIPIcs.CCC.2019.25,
  author =	{Coudron, Matthew and Slofstra, William},
  title =	{{Complexity Lower Bounds for Computing the Approximately-Commuting Operator Value of Non-Local Games to High Precision}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.25},
  URN =		{urn:nbn:de:0030-drops-108478},
  doi =		{10.4230/LIPIcs.CCC.2019.25},
  annote =	{Keywords: Quantum complexity theory, Non-local game, Multi-prover interactive proof, Entanglement}
}
Document
Tight SoS-Degree Bounds for Approximate Nash Equilibria

Authors: Aram Harrow, Anand V. Natarajan, and Xiaodi Wu

Published in: LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)


Abstract
Nash equilibria always exist, but are widely conjectured to require time to find that is exponential in the number of strategies, even for two-player games. By contrast, a simple quasi-polynomial time algorithm, due to Lipton, Markakis and Mehta (LMM), can find approximate Nash equilibria, in which no player can improve their utility by more than epsilon by changing their strategy. The LMM algorithm can also be used to find an approximate Nash equilibrium with near-maximal total welfare. Matching hardness results for this optimization problem re found assuming the hardness of the planted-clique problem (by Hazan and Krauthgamer) and assuming the Exponential Time Hypothesis (by Braverman, Ko and Weinstein). In this paper we consider the application of the sum-squares (SoS) algorithm from convex optimization to the problem of optimizing over Nash equilibria. We show the first unconditional lower bounds on the number of levels of SoS needed to achieve a constant factor approximation to this problem. While it may seem that Nash equilibria do not naturally lend themselves to convex optimization, we also describe a simple LP (linear programming) hierarchy that can find an approximate Nash equilibrium in time comparable to that of the LMM algorithm, although neither algorithm is obviously a generalization of the other. This LP can be viewed as arising from the SoS algorithm at log(n) levels - matching our lower bounds. The lower bounds involve a modification of the Braverman-Ko-Weinstein embedding of CSPs into strategic games and techniques from sum-of-squares proof systems. The upper bound (i.e. analysis of the LP) uses information-theory techniques that have been recently applied to other linear- and semidefinite-programming hierarchies.

Cite as

Aram Harrow, Anand V. Natarajan, and Xiaodi Wu. Tight SoS-Degree Bounds for Approximate Nash Equilibria. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 22:1-22:25, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{harrow_et_al:LIPIcs.CCC.2016.22,
  author =	{Harrow, Aram and Natarajan, Anand V. and Wu, Xiaodi},
  title =	{{Tight SoS-Degree Bounds for Approximate Nash Equilibria}},
  booktitle =	{31st Conference on Computational Complexity (CCC 2016)},
  pages =	{22:1--22:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-008-8},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{50},
  editor =	{Raz, Ran},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.22},
  URN =		{urn:nbn:de:0030-drops-58565},
  doi =		{10.4230/LIPIcs.CCC.2016.22},
  annote =	{Keywords: Approximate Nash Equilibrium, Sum of Squares, LP, SDP}
}
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