This academic paper explores methods to improve Bayesian optimization (BO), a process for finding optimal settings for complex, costly functions. The authors address the challenge of maximizing acquisition functions, which are heuristics guiding BO's search and are often difficult to optimize, especially when evaluating multiple points simultaneously. They demonstrate that Monte Carlo integration allows for gradient-based optimization of a wide range of acquisition functions. Furthermore, they identify a family of acquisition functions that possess submodular properties, making them suitable for efficient greedy maximization. Experimental results indicate that these techniques significantly enhance BO performance, particularly as the complexity of the optimization increases.