Tight SoS-Degree Bounds for Approximate Nash Equilibria

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 ε 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 were 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

Verifying quantum proofs with entangled games

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Physics, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 197-205).A team of students has been given a challenging physics exam: find the ground energy of a complicated, n-spin system. Even if they succeed, how can the examiners be sure that their answer is correct without physically measuring all n spins of the ground state, or worse, having to read a description of the 2n components of its wavefunction? The main result of this thesis is a protocol such that, if the examiners are allowed to separately interrogate multiple students, they can be confident that the students possess the n-spin ground state as well as learn its energy to high precision, after exchanging just O(log(n)) bits of classical communication with the students! The protocol and its analysis combine classical computer science techniques for efficiently checking proofs with Bell inequalities. Stated more formally, the main result of this thesis is a multi-prover interactive proof protocol, in which a classical verifier exchanging only O(log(n)) bits of classical communication with 7 untrusted, entangled provers can certify that they share between them an encoding of an n-qubit quantum state Ib), and estimate its energy under a local Hamiltonian H to high (1/ poly(n)) precision. As a consequence, we show that, under poly-time randomized reductions, it is QMA-hard to estimate the entangled value of a nonlocal game up to constant error, proving the quantum entangled games PCP conjecture of Fitzsimons and Vidick. Our main technical innovations are two constructions of robust self-tests for entanglement: two-player nonlocal games where to succeed with probability E-close to 1, the players must share a state that is [delta] = poly([epsilon])-close in trace distance to n EPR pairs. These tests are robust in that [delta] is independent of the number n of EPR pairs being tested. Our techniques draw heavily on the original, "algebraic" proof of the PCP theorem in classical complexity theory, and in particular, each of our robust self-tests is based on a classical locally-testable error correcting code: the first on the Hadamard code and the associated linearity test of Blum, Luby, and Rubinfeld, and the second on Reed-Muller code and the associated low-degree test of Raz and Safra.Supported by NSF Grant CCF-1629809 ARO Contract Number W911NF-12-0486 NSF CAREER Grant CCF-1452616by Anand Venkat Natarajan.Ph. D