Approximate message passing (AMP) is a family of iterative algorithms that generalize matrix power iteration. AMP algorithms are known to optimally solve many average-case optimization problems. In this paper, we show that a large class of AMP algorithms can be simulated in polynomial time by \emph{local statistics hierarchy} semidefinite programs (SDPs), even when an unknown principal minor of measure $1/\mathrm{polylog}(\mathrm{dimension})$ is adversarially corrupted. Ours are the first robust guarantees for many of these problems. Further, our results offer an interesting counterpoint to strong lower bounds against less constrained SDP relaxations for average-case max-cut-gain (a.k.a. "optimizing the Sherrington-Kirkpatrick Hamiltonian") and other problems.

翻译：近似消息传递（AMP）是一类推广矩阵幂迭代的迭代算法。已知AMP算法能在许多平均情形优化问题中达到最优解。本文证明，即使存在一个测度为$1/\mathrm{polylog}(\mathrm{维度})$的未知主子式遭受对抗性破坏，一大类AMP算法仍可通过*局部统计层级*半定规划（SDP）在多项式时间内被模拟。这是针对许多此类问题的首批稳健性保证。此外，我们的结果为平均情形最大割增益（即"优化夏莫林顿-柯克帕特里克哈密顿量"）等问题中，较不具约束性的SDP松弛所面临的强下界提供了一个有趣的对比。