The Maximum k-plex Problem is an important combinatorial optimization problem with increasingly wide applications. Due to its exponential time complexity, many heuristic methods have been proposed which can return a good-quality solution in a reasonable time. However, most of the heuristic algorithms are memoryless and unable to utilize the experience during the search. Inspired by the multi-armed bandit (MAB) problem in reinforcement learning (RL), we propose a novel perturbation mechanism named BLP, which can learn online to select a good vertex for perturbation when getting stuck in local optima. To our best of knowledge, this is the first attempt to combine local search with RL for the maximum $ k $-plex problem. Besides, we also propose a novel strategy, named Dynamic-threshold Configuration Checking (DTCC), which extends the original Configuration Checking (CC) strategy from two aspects. Based on the BLP and DTCC, we develop a local search algorithm named BDCC and improve it by a hyperheuristic strategy. The experimental result shows that our algorithms dominate on the standard DIMACS and BHOSLIB benchmarks and achieve state-of-the-art performance on massive graphs.