Efficient Lipschitzian Global Optimization of Hölder Continuous Multivariate Functions
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This study presents an effective global optimization technique designed for multivariate functions that are Hölder continuous. Unlike traditional methods that construct lower bounding proxy functions, this algorithm employs a predetermined query creation rule that makes it computationally superior. The algorithm's performance is assessed using the average or cumulative regret, which also implies a bound for the simple regret and reflects the overall effectiveness of the approach. The results show that with appropriate parameters the algorithm attains an average regret bound of O(T^-α/n) for optimizing a Hölder continuous target function with Hölder exponent α in an n-dimensional space within a given time horizon T. We demonstrate that this bound is minimax optimal.
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