We study the problem of algorithmically optimizing the Hamiltonian $H_N$ of a spherical or Ising mixed $p$-spin glass. The maximum asymptotic value $\mathsf{OPT}$ 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 $\mathsf{ALG}$ 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, $\mathsf{ALG} < \mathsf{OPT}$ can also occur, and no efficient algorithm producing an objective value exceeding $\mathsf{ALG}$ is known. We prove that for mixed even $p$-spin models, no algorithm satisfying an overlap concentration property can produce an objective larger than $\mathsf{ALG}$ 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 $\mathsf{ALG}$ 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.
翻译:我们从逻辑上研究如何优化汉密尔顿 $H_N美元这一球状或球状混合 $p1 的玻璃。 最大无光值为$H_N/ OPT}$H_ N美元, 其特点是被称为巴黎公式的变异原则, 由塔拉格兰首次证明, 由潘琴科更笼统地证明。 最近开发了近似的信息传递算法, 将美元/ N美元有效优化到一个值为$mathsf{ ALG} 。 由扩展的巴黎公式提供的, 将功能顺序参数的更大空间最小化。 这两个目标对旋转眼镜显示无重叠属性的相同。 然而, 美元\ mathsf{ ALG} < gmathsf{OPT} 也可能发生, 而没有有效的算法产生超过$mathfsf{ ALG} 的客观值。 我们证明, 即使是混合的 美元- spinal化的算法框架, 我们无法通过一个重叠的浓度属性生成比 $\mathal f calalalalalalalal ralal rudeal rudeal rudeal sult exal exal sults the slations the slations slations slations slations slations slations slations slevation.