## 4 Hyperparameter Optimization
#### The Computational Complexity and Resource Usage  
In the present study, both classical path-planning methods (A*, RRT*, GA, and PSO) and off-policy Deep Reinforcement Learning (DRL) algorithms (SAC, TD3, and TQC) were investigated within the same high-fidelity AUV environment. Although the primary metric reported was the execution time for each method under a fixed scenario, it is equally important to consider how these approaches scale as the problem dimensions grow or when the environment becomes more cluttered. From a theoretical standpoint, grid-based methods such as A* may exhibit time complexity in the order of \(O(n^{2}\log n)\) if an \(n\times n\) map is employed, with the memory usage driven by the need to maintain a priority queue, as well as a closed list of the visited nodes. Sampling-based approaches like the RRT* typically have a time complexity \(O(k\log k)\), where \(k\) is the number of sampled nodes. While each new sample is relatively cheap to generate, large numbers of samples are often required in environments with narrow passages or complex obstacles, which can drive up both the time and memory costs. Genetic and swarm-based methods (GA, PSO) can be more difficult to characterize precisely; in each generation, a population of candidate solutions is evaluated, meaning the time cost scales with the population size and the cost of collision checks. Although these checks can be performed on a standard CPU, the computational burden may still become substantial for large maps. By contrast, off-policy DRL algorithms such as SAC, TD3, and TQC, although shown to converge quickly under the training procedure described in this paper, impose a higher computational load during the learning phase. Larger neural networks (e.g., hidden layers with a size of 1024 or more) require many floating-point operations per gradient step, and the replay buffer, typically storing up to one million transitions, can drive up the memory usage. Distributional or multi-critic extensions like TQCs add further overhead by maintaining additional parameters and quantile estimates. To manage this scale effectively, a GPU is usually essential for accelerating the neural network updates and ensuring that training completes within a practical timescale. After training, however, inference can be run at low costs, making this policy suitable for near-real-time replanning or adaptation once an AUV has been deployed. In more extensive or dynamically changing environments, the classical planners may require either downscaling the map resolution (which can compromise accuracy) or increasingly large searches or population sizes, thus inflating the time and memory consumption. DRL, on the other hand, may call for more complex network architectures to model the richer state space (e.g., sonar readings or multi-layer bathymetry), and the replay buffer may need to become even larger to capture the diverse experience. Although this translates into a substantial one-time training cost, the final policy can then provide immediate action decisions, which is advantageous for missions that require frequent online adjustments. Table 4 illustrates the approximate time complexity and hardware demands for each algorithm category, aligning with the problem setup in this paper. The classical methods are generally less memory-hungry and do not strictly mandate the GPU usage. Nonetheless, their runtimes can grow significantly with higher map resolutions or complex obstacle layouts. Meanwhile, DRL approaches typically require stronger hardware resources during training but achieve rapid policy inference. Overall, the choice between a classical planner and a learned DRL policy depends not only on the accuracy and the mission time but also on practical factors such as the available memory and GPU capabilities and whether the scenario allows for offline training or demands immediate in situ adaptation.