Plug-and-Play Alternating Direction Method of Multipliers (PnP-ADMM) is a
widely-used algorithm for solving inverse problems by integrating physical
measurement models and convolutional neural network (CNN) priors. PnP-ADMM has
been theoretically proven to converge for convex data-fidelity terms and
nonexpansive CNNs. It has however been observed that PnP-ADMM often empirically
converges even for expansive CNNs. This paper presents a theoretical
explanation for the observed stability of PnP-ADMM based on the interpretation
of the CNN prior as a minimum mean-squared error (MMSE) denoiser. Our
explanation parallels a similar argument recently made for the iterative
shrinkage/thresholding algorithm variant of PnP (PnP-ISTA) and relies on the
connection between MMSE denoisers and proximal operators. We also numerically
evaluate the performance gap between PnP-ADMM using a nonexpansive DnCNN
denoiser and expansive DRUNet denoiser, thus motivating the use of expansive
CNNs.

基于解释卷积神经网络先验为最小均方误差（MMSE）降噪器的理论解析，论文对 PnP-ADMM 的稳定性进行了解释，并通过对非扩展 DnCNN 降噪器和扩展 DRUNet 降噪器之间性能差距的数值评估，进一步验证了使用扩展卷积神经网络的动机。

非凸 PnP-ADMM 与 MMSE 去噪算法的收敛性

Convergence of Nonconvex PnP-ADMM with MMSE Denoisers

Proximal operators are ubiquitous in inverse problems, commonly appearing as
part of algorithmic strategies to regularize problems that are otherwise
ill-posed. Modern deep learning models have been brought to bear for these
tasks too, as in the framework of plug-and-play or deep unrolling, where they
loosely resemble proximal operators. Yet, something essential is lost in
employing these purely data-driven approaches: there is no guarantee that a
general deep network represents the proximal operator of any function, nor is
there any characterization of the function for which the network might provide
some approximate proximal. This not only makes guaranteeing convergence of
iterative schemes challenging but, more fundamentally, complicates the analysis
of what has been learned by these networks about their training data. Herein we
provide a framework to develop learned proximal networks (LPN), prove that they
provide exact proximal operators for a data-driven nonconvex regularizer, and
show how a new training strategy, dubbed proximal matching, provably promotes
the recovery of the log-prior of the true data distribution. Such LPN provide
general, unsupervised, expressive proximal operators that can be used for
general inverse problems with convergence guarantees. We illustrate our results
in a series of cases of increasing complexity, demonstrating that these models
not only result in state-of-the-art performance, but provide a window into the
resulting priors learned from data.

学习的近似算子网络提供一种通用的、无监督的、表达力强的近似算子，可用于具有收敛性保证的一般逆问题。

学习到的近端网络用于逆问题中的先验知识

What's in a Prior? Learned Proximal Networks for Inverse Problems

In this paper, we introduce various mechanisms to obtain accelerated
first-order stochastic optimization algorithms when the objective function is
convex or strongly convex. Specifically, we extend the Catalyst approach
originally designed for deterministic objectives to the stochastic setting.
Given an optimization method with mild convergence guarantees for strongly
convex problems, the challenge is to accelerate convergence to a
noise-dominated region, and then achieve convergence with an optimal worst-case
complexity depending on the noise variance of the gradients. A side
contribution of our work is also a generic analysis that can handle inexact
proximal operators, providing new insights about the robustness of stochastic
algorithms when the proximal operator cannot be exactly computed.

本文介绍了在目标函数为凸或强凸函数时获取加速一阶随机优化算法的各种机制，同时扩展了最初用于确定性目标的 Catalyst 方法到随机问题领域，并提供了一个新的关于处理不精确近端算子时的鲁棒性的泛化分析

随机组合优化的通用加速框架

A Generic Acceleration Framework for Stochastic Composite Optimization

In this paper, we set forth a new vision of reinforcement learning developed
by us over the past few years, one that yields mathematically rigorous
solutions to longstanding important questions that have remained unresolved:
(i) how to design reliable, convergent, and robust reinforcement learning
algorithms (ii) how to guarantee that reinforcement learning satisfies
pre-specified "safety" guarantees, and remains in a stable region of the
parameter space (iii) how to design "off-policy" temporal difference learning
algorithms in a reliable and stable manner, and finally (iv) how to integrate
the study of reinforcement learning into the rich theory of stochastic
optimization. In this paper, we provide detailed answers to all these questions
using the powerful framework of proximal operators.
The key idea that emerges is the use of primal dual spaces connected through
the use of a Legendre transform. This allows temporal difference updates to
occur in dual spaces, allowing a variety of important technical advantages. The
Legendre transform elegantly generalizes past algorithms for solving
reinforcement learning problems, such as natural gradient methods, which we
show relate closely to the previously unconnected framework of mirror descent
methods. Equally importantly, proximal operator theory enables the systematic
development of operator splitting methods that show how to safely and reliably
decompose complex products of gradients that occur in recent variants of
gradient-based temporal difference learning. This key technical innovation
makes it possible to finally design "true" stochastic gradient methods for
reinforcement learning. Finally, Legendre transforms enable a variety of other
benefits, including modeling sparsity and domain geometry. Our work builds
extensively on recent work on the convergence of saddle-point algorithms, and
on the theory of monotone operators.

本文提出了一种新的加强学习算法，使用似然变换、原始 - 对偶空间以及近端算子等技术，解决了关于如何设计可靠、收敛、稳健的加强学习算法、如何保证算法具有预设的安全保证、如何稳定地设计离线 - 在线学习算法、如何将加强学习与随机优化理论相结合等问题，为相关理论做出了丰富的探索和创新。