# Deep Learning-Based Step Size Determination for Hill Climbing Metaheuristics
## 1 Introduction  
Metaheuristics are powerful optimization methods known for their versatility and adaptability to diverse problem types, ranging from non-linear and non-convex challenges [1] to large-scale, high-dimensional tasks [2]. The stochastic nature of metaheuristics raises innovative solutions, while their scalability and parallelization potential [3] enable efficiency in large computations. Additionally, their simplicity and ease of implementation make them generally applicable, yet they remain flexible enough to incorporate domain-specific heuristics or hybrid approaches for tailored problem-solving. These benefits have made metaheuristics a popular choice across several fields [4]. Although Machine Learning (ML) methods have become increasingly popular in the last decade [5, 6], they have been overshadowed in research on metaheuristics. The principles of different metaheuristic search algorithms are generally straightforward, but these usually have several parameters. This is not a shortcoming that can be eliminated because it gives them their flexibility. For example, these parameters make it possible to find the correct balance between the exploration and exploitation phases, control the speed of convergence, and so on. Even the simplest Hill Climbing (HC) algorithm needs several parameters (staring positions, step size, and number of examined directions) and control functions (fitness calculation and stopping conditions). These parameters are usually domain-specific and based on the attributes of the search space. Therefore, their setting requires considerable experience or a time-consuming trial-and-error process to achieve the most effective results. The novel idea of our research is to use ML to enhance the already existing metaheuristics to make them capable of learning these parameters themselves. There are several possibilities to build these hybrid methods, for example, to design a Tabu Search [7] using ML to identify the already visited locations or to use ML to define the stopping conditions of the main iteration, answering the question of whether it is worth running further iterations of a local search or whether it is better to restart it from another starting point. The design of these hybrid methods is very computationally intensive, as a large number of heuristics needs to be run to generate the training database, followed by another resource intensive neural network training; and finally, a lot more benchmark executions are required to ensure reliable evaluation (and this may need to be repeated when testing different ML models). As a preliminary step in this direction, this paper presents the idea of a hybrid ML-assisted HC method capable of determining the appropriate step size dynamically, based on the knowledge extracted from previous runs. [PERSON] et al. [8] presented a novel method based on variable step size, solving a problem in the domain of tidal energy application with a maximum power tracking control technique for a power generator system. Another example can be the work of [PERSON] et al. [9], which combines the attributes of individual searches and population-based algorithms. [PERSON] also presented [10], a novel population-based adaptation scheme with a self-adaptive step size based on the distance of the current position and a randomly selected sample of the population during each iteration. [PERSON] et al. have multiple papers on the topic of the HC method using a random step size [11; 12]. These domain-based methods can be very efficient, but they all need the knowledge of the attributes of the search space. In the case of complex problems, it can be very challenging to find the appropriate adaptations. The novel idea of this paper is that this domain-specific knowledge can be extracted via ML. This paper presents some details about the HC algorithm and its variants, followed by the novel hybrid approach, detailing the methodology behind generating training data, refining the algorithm, and evaluating its effectiveness.