Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses

10/15/2021
by   Brice Huang, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/25/2020

Low-Degree Hardness of Random Optimization Problems

We consider the problem of finding nearly optimal solutions of optimizat...
research
03/21/2023

Algorithmic Threshold for Multi-Species Spherical Spin Glasses

We study efficient optimization of the Hamiltonians of multi-species sph...
research
08/01/2021

The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures

The problem of optimizing over random structures emerges in many areas o...
research
06/03/2021

The Algorithmic Phase Transition of Random k-SAT for Low Degree Polynomials

Let Φ be a uniformly random k-SAT formula with n variables and m clauses...
research
10/06/2022

Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

We study random constraint satisfaction problems (CSPs) in the unsatisfi...
research
03/01/2022

On Orthogonal Approximate Message Passing

Approximate Message Passing (AMP) is an efficient iterative parameter-es...

Please sign up or login with your details

Forgot password? Click here to reset