This paper presents a unified framework for constructing Approximate Message Passing (AMP) algorithms for rotationally-invariant models. By employing a general iterative algorithm template and reducing it to long-memory Orthogonal AMP (OAMP), we systematically derive the correct Onsager terms of AMP algorithms. This approach allows us to rederive an AMP algorithm introduced by Fan and Opper et al., while shedding new light on the role of free cumulants of the spectral law. The free cumulants arise naturally from a recursive centering operation, potentially of independent interest beyond the scope of AMP. To illustrate the flexibility of our framework, we introduce two novel AMP variants and apply them to estimation in spiked models.

The exploding popularity of online content and its user base poses an evermore challenging matching problem for modern recommendation systems. Unlike other frontiers of machine learning such as natural language, recommendation systems are responsible for collecting their own data. Simply exploiting current knowledge can lead to pernicious feedback loops but naive exploration can detract from user experience and lead to reduced engagement. This exploration-exploitation trade-off is exemplified in the classic multi-armed bandit problem for which algorithms such as upper confidence bounds (UCB) and Thompson sampling (TS) demonstrate effective performance. However, there have been many challenges to scaling these approaches to settings which do not exhibit a conjugate prior structure. Recent scalable approaches to uncertainty quantification via epinets have enabled efficient approximations of Thompson sampling even when the learning model is a complex neural network. In this paper, we demonstrate the first application of epinets to an online recommendation system. Our experiments demonstrate improvements in both user traffic and engagement efficiency on the Facebook Reels online video platform.

We study the problem of estimating a rank one signal matrix from an observed matrix generated by corrupting the signal with additive rotationally invariant noise. We develop a new class of approximate message-passing algorithms for this problem and provide a simple and concise characterization of their dynamics in the high-dimensional limit. At each iteration, these algorithms exploit prior knowledge about the noise structure by applying a non-linear matrix denoiser to the eigenvalues of the observed matrix and prior information regarding the signal structure by applying a non-linear iterate denoiser to the previous iterates generated by the algorithm. We exploit our result on the dynamics of these algorithms to derive the optimal choices for the matrix and iterate denoisers. We show that the resulting algorithm achieves the smallest possible asymptotic estimation error among a broad class of iterative algorithms under a fixed iteration budget.