论文标题
Sudakov-Fernique post-amp，以及自来水能源本地凸的新证明
Sudakov-Fernique post-AMP, and a new proof of the local convexity of the TAP free energy
论文作者
论文摘要
在现代统计和机器学习中的许多问题中，人们通常是为了确定非凸风险功能上的一阶方法最终进入了风险是本地凸的参数空间区域。我们得出了一种渐近比较不平等，我们称之为Sudakov-Fernique后AMP不平等现象，在涉及GOE矩阵的一定类别的问题中，它能够探测优化景观在近似消息传递（AMP）algorithm的迭代周围的优化景观的特性。作为其使用的一个例子，我们提供了一个新的，可以说是简单的证据，证明了Celentano等人的某些结果。 （2021），确定在$ \ mathbb {z} _2 $ -Synchronization问题中所谓的Tap自由能在其收敛到的区域中是本地凸。我们进一步证明了El Alaoui等人的猜想。 （2022）涉及相关但独特的自由能的局部凸度，结果证实了它们的算法有效地从Sherrington-Kirkpatrick Gibbs中进行了整个“ Easy”状态的测量。
In many problems in modern statistics and machine learning, it is often of interest to establish that a first order method on a non-convex risk function eventually enters a region of parameter space in which the risk is locally convex. We derive an asymptotic comparison inequality, which we call the Sudakov-Fernique post-AMP inequality, which, in a certain class of problems involving a GOE matrix, is able to probe properties of an optimization landscape locally around the iterates of an approximate message passing (AMP) algorithm. As an example of its use, we provide a new, and arguably simpler, proof of some of the results of Celentano et al. (2021), which establishes that the so-called TAP free energy in the $\mathbb{Z}_2$-synchronization problem is locally convex in the region to which AMP converges. We further prove a conjecture of El Alaoui et al. (2022) involving the local convexity of a related but distinct TAP free energy, which, as a consequence, confirms that their algorithm efficiently samples from the Sherrington-Kirkpatrick Gibbs measure throughout the "easy" regime.
