论文标题
高维反问题中正则参数的最大似然估计：一种经验贝叶斯方法。第二部分：理论分析
Maximum likelihood estimation of regularisation parameters in high-dimensional inverse problems: an empirical Bayesian approach. Part II: Theoretical Analysis
论文作者
论文摘要
本文对我们的同伴论文[49]中提出的三个随机近似近端梯度算法进行了详细的理论分析，以通过边际最大似然估计来设置正则化参数。我们证明了更通用的随机近似方案的收敛性，该方案包括[49]作为特殊情况的三种算法。这包括具有自然且易于验证的条件以及收敛速率的明确界限，包括渐近和非反应收敛结果。重要的是，该理论也很普遍，因为它可以应用于其他棘手的优化问题。这项工作的主要新颖性是我们方案的随机梯度估计值是由马尔可夫近端链蒙特卡洛采样器构建的。这允许使用有效地扩展到大型问题并为我们提供精确理论保证的采样器。
This paper presents a detailed theoretical analysis of the three stochastic approximation proximal gradient algorithms proposed in our companion paper [49] to set regularization parameters by marginal maximum likelihood estimation. We prove the convergence of a more general stochastic approximation scheme that includes the three algorithms of [49] as special cases. This includes asymptotic and non-asymptotic convergence results with natural and easily verifiable conditions, as well as explicit bounds on the convergence rates. Importantly, the theory is also general in that it can be applied to other intractable optimisation problems. A main novelty of the work is that the stochastic gradient estimates of our scheme are constructed from inexact proximal Markov chain Monte Carlo samplers. This allows the use of samplers that scale efficiently to large problems and for which we have precise theoretical guarantees.
