Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses
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We study the problem of algorithmically optimizing the Hamiltonian H_N of a spherical or Ising mixed p-spin glass. The maximum asymptotic value 𝖮𝖯𝖳 of H_N/N is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing algorithms efficiently optimize H_N/N up to a value 𝖠𝖫𝖦 given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overlap gap property. However, 𝖠𝖫𝖦 < 𝖮𝖯𝖳 can also occur, and no efficient algorithm producing an objective value exceeding 𝖠𝖫𝖦 is known. We prove that for mixed even p-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than 𝖠𝖫𝖦 with non-negligible probability. This property holds for all algorithms with suitably Lipschitz dependence on the disorder coefficients of H_N. It encompasses natural formulations of gradient descent, approximate message passing, and Langevin dynamics run for bounded time and in particular includes the algorithms achieving 𝖠𝖫𝖦 mentioned above. To prove this result, we substantially generalize the overlap gap property framework introduced by Gamarnik and Sudan to arbitrary ultrametric forbidden structures of solutions.
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