Projected Gaussian Markov Improvement Algorithm for High-Dimensional Discrete Optimization via Simulation
提出pGMIA算法，通过将解空间投影到低维空间来降低高斯马尔可夫随机场推断成本，解决高维离散优化问题，并证明其渐近收敛性。
This article considers a discrete optimization via simulation (DOvS) problem defined on a graph embedded in the high-dimensional integer grid. Several DOvS algorithms that model the responses at the solutions as a realization of a Gaussian Markov random field (GMRF) have been proposed exploiting its inferential power and computational benefits. However, the computational cost of inference increases exponentially in dimension. We propose the projected Gaussian Markov improvement algorithm (pGMIA), which projects the solution space onto a lower-dimensional space creating the region-layer graph to reduce the cost of inference. Each node on the region-layer graph can be mapped to a set of solutions projected to the node; these solutions form a lower-dimensional solution-layer graph. We define the response at each region-layer node to be the average of the responses within the corresponding solution-layer graph. From this relation, we derive the region-layer GMRF to model the region-layer responses. The pGMIA alternates between the two layers to make a sampling decision at each iteration. It first selects a region-layer node based on the lower-resolution inference provided by the region-layer GMRF, then makes a sampling decision among the solutions within the solution-layer graph of the node based on the higher-resolution inference from the solution-layer GMRF. To solve even higher-dimensional problems (e.g., 100 dimensions), we also propose the pGMIA+: a multi-layer extension of the pGMIA. We show that both pGMIA and pGMIA+ converge to the optimum almost surely asymptotically and empirically demonstrate their competitiveness against state-of-the-art high-dimensional Bayesian optimization algorithms.