统计推断的近似调查传播
Approximate Survey Propagation for Statistical Inference
摘要：
近似消息传递算法在过去的十年中受到了广泛的关注。在本文中，我们介绍了一个变体的AMP算法，考虑到玻璃的性质所考虑的系统。我们把这种算法称为近似调查传播（ASP），并推导出它的一类低秩矩阵估计问题。我们推导了ASP算法的状态演化，并证明了它再现了无序系统物理中著名的一步复本对称性破缺（1RSB）不动点方程。因此，我们的推导给出了一个具体的算法意义的1RSB方程是独立的利益。我们的ASP的性能的收敛性和均方误差作为一个功能的自由Parisi参数s。我们的结论是，当有一个真正的生成模型和推理模型之间的模型不匹配，AMP的性能迅速下降的MSE和收敛，而ASP收敛在一个较大的制度，可以达到较低的错误。除其他结果外，我们的分析使我们得出了一个惊人的假设，即只要s（或其他参数）可以以恢复Nishimori条件$M=Q>0$的方式设置，那么相应的算法就能够达到均方误差低至模型及其参数已知且在推理过程中完全匹配时获得的贝叶斯最优误差。
英文摘要：
Approximate message passing algorithm enjoyed considerable attention in the last decade. In this paper we introduce a variant of the AMP algorithm that takes into account glassy nature of the system under consideration. We coin this algorithm as the approximate survey propagation (ASP) and derive it for a class of low-rank matrix estimation problems. We derive the state evolution for the ASP algorithm and prove that it reproduces the one-step replica symmetry breaking (1RSB) fixed-point equations, well-known in physics of disordered systems. Our derivation thus gives a concrete algorithmic meaning to the 1RSB equations that is of independent interest. We characterize the performance of ASP in terms of convergence and mean-squared error as a function of the free Parisi parameter s. We conclude that when there is a model mismatch between the true generative model and the inference model, the performance of AMP rapidly degrades both in terms of MSE and of convergence, while ASP converges in a larger regime and can reach lower errors. Among other results, our analysis leads us to a striking hypothesis that whenever s (or other parameters) can be set in such a way that the Nishimori condition $M=Q>0$ is restored, then the corresponding algorithm is able to reach mean-squared error as low as the Bayes-optimal error obtained when the model and its parameters are known and exactly matched in the inference procedure.
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