A Bi-level Optimization Algorithm Based On Multi-task Bayesian Optimization

Bi-level optimization has wide applications in reality,such as automatic machine learning,traffic planning,industrial design,etc.Multi-task Bayesian optimization,as an efficient black-box function optimization method,utilizes the correlation information between tasks to improve the convergence speed of optimization.Bi-level optimization is a kind of optimization problem with a nested structure,where the objective function itself contains the solution to another optimization problem.Therefore,every time the objective function is evaluated,the internal optimization problem needs to be solved,and its solution is used to update the parameters of the objective function.Although methods using evolutionary computation has multiple advantages in recent years,the characteristics of evolutionary computation lead to a large number of function evaluations.Therefore,when dealing with real optimization problems,it is necessary to further improve the optimization efficiency and reduce the actual number of function evaluations.To address this,this paper proposes a bi-level optimization algorithm based on multi-task Bayesian optimization.By modeling the historical data together,the algorithm significantly reduces the number of actual objective function called in the upper and lower levels,and has high feasibility and effectiveness while maintaining optimization accuracy.The exact summary is as follows:(1)A new approach has been proposed to improve efficiency in bi-level optimization by fully utilizing historical evaluation data.When solving bi-level optimization problems,the lower level optimization is typically regarded as the inner loop of the upper-level optimization to ensure optimality of the lower-level variables.Therefore,a complete lower level optimization is required every time the upper level function is evaluated,resulting in a large number of calls to the lower-level objective function.However,other algorithms do not reuse these historical data,which is an extremely inefficient waste of resources.The algorithm proposed in this article stores these historical data and reuses them based on the Euclidean distance as a condition for utilization.Although it increases memory consumption,it greatly reduces the number of function evaluations through modeling among multiple tasks.(2)A bi-level optimization algorithm based on multi-task Bayesian optimization is proposed.According to the continuity principle,the values of functions between adjacent points should also be highly similar.From this point,this paper proposes a method that can use multi-task Bayesian optimization to optimize the two-layer optimization process and improve efficiency.The algorithm proposed in the paper stores historical data and takes them out when conditions are met,and establishes a multi-task Gaussian process to model them together.It is experimentally demonstrated that this method can significantly reduce the number of function evaluations with similar optimization accuracy.(3)An improved Bayesian optimization algorithm which dynamically adjusts the search region is proposed.When dealing with optimization problems,there may be cases where the span of function values in orders of magnitude is very large.To solve this problem,this paper proposes a method using dynamically adjusted optimization modeling regions.The method uses the hyperparameter length scale 7)in the Gaussian model as a basis for dynamically adjusting the search region.According to the experimental demonstration,it not only solves this problem,but also further improves the optimization convergence speed.