This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularised by Moreau-Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretisation of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretisation is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is $\kappa$-strongly log-concave (i.e., requiring in the order of $\sqrt{\kappa}$ iterations to converge, similarly to accelerated optimisation schemes), comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13, 2 (2020), pp. 905-935] which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin diffusion targeting a Moreau-Yosida approximation of the posterior distribution of interest, and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For targets that are $\kappa$-strongly log-concave, the provided non-asymptotic convergence analysis also identifies the optimal time step which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise, with assumption-driven and data-driven convex priors.

翻译：本文提出了一种新的加速近端马尔可夫链蒙特卡洛方法，用于在具有凸几何结构的成像反问题中执行贝叶斯推断。该策略采用随机松弛近端点迭代形式，具有两种互补解释。对于光滑或通过Moreau-Yosida平滑正则化的模型，该算法等价于针对目标后验分布的过阻尼Langevin扩散的隐式中点离散化。这种离散化对高斯目标渐近无偏，并且对于任何$\kappa$-强对数凹目标（即需约$\sqrt{\kappa}$次迭代收敛，与加速优化方案类似），其收敛速度优于[M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13, 2 (2020), pp. 905-935]方法——后者仅对高斯目标具有可证明的加速性且存在偏差。对于非光滑模型，该算法等价于针对目标后验分布的Moreau-Yosida近似的Langevin扩散的Leimkuhler-Matthews离散化，因此比基于Euler-Maruyama离散化的常规未调整Langevin策略实现显著更低的偏差。对于$\kappa$-强对数凹目标，所提供的非渐近收敛分析还确定了最大化收敛速度的最优时间步长。通过一系列涉及高斯噪声和泊松噪声的图像去卷积实验（采用假设驱动和数据驱动的凸先验），验证了所提方法的有效性。