This study proposes a trainable sampling-based solver for combinatorial
optimization problems (COPs) using a deep-learning technique called deep
unfolding. The proposed solver is based on the Ohzeki method that combines
Markov-chain Monte-Carlo (MCMC) and gradient descent, and its step sizes are
trained by minimizing a loss function. In the training process, we propose a
sampling-based gradient estimation that substitutes auto-differentiation with a
variance estimation, thereby circumventing the failure of back propagation due
to the non-differentiability of MCMC. The numerical results for a few COPs
demonstrated that the proposed solver significantly accelerated the convergence
speed compared with the original Ohzeki method.

该研究提出了一种可训练的基于采样的求解器用于组合优化问题，使用一种称为深展开的深度学习技术。该求解器基于 Ohzeki 方法，结合了马尔可夫链蒙特卡洛（MCMC）和梯度下降，并通过最小化损失函数来训练其步长。在训练过程中，我们提出了一种基于采样的梯度估计，用方差估计代替自动微分，从而避免了由于 MCMC 的不可微分性导致的反向传播失败。对于几个组合优化问题的数值结果表明，与原始的 Ohzeki 方法相比，所提出的求解器显著加速了收敛速度。

基于深度展开的马尔可夫链蒙特卡洛梯度下降的收敛加速

Convergence Acceleration of Markov Chain Monte Carlo-based Gradient  Descent by Deep Unfolding

Estimation of parameters of a diffusion based on discrete time observations
poses a difficult problem due to the lack of a closed form expression for the
likelihood. From a Bayesian computational perspective it can be casted as a
missing data problem where the diffusion bridges in between discrete-time
observations are missing. The computational problem can then be dealt with
using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown
parameters appear in the diffusion coefficient, direct implementation of
data-augmentation results in a Markov chain that is reducible. Furthermore,
data-augmentation requires efficient sampling of diffusion bridges, which can
be difficult, especially in the multidimensional case.
We present a general framework to deal with with these problems that does not
rely on discretisation. The construction generalises previous approaches and
sheds light on the assumptions necessary to make these approaches work. We
define a random-walk type Metropolis-Hastings sampler for updating diffusion
bridges. Our methods are illustrated using guided proposals for sampling
diffusion bridges. These are Markov processes obtained by adding a guiding term
to the drift of the diffusion. We give general guidelines on the construction
of these proposals and introduce a time change and scaling of the guided
proposal that reduces discretisation error. Numerical examples demonstrate the
performance of our methods.

本文提出了一种新的框架来解决扩散过程参数估计中存在的难题，并将其应用于 Markov 链蒙特卡罗法中的数据扩充。该方法使用了一种基于随机游走的 Metropolis-Hastings 采样器来更新扩散程的参数，成功地解决了先前方法中的困难点。