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One of the most critical problems in machine learning is HyperParameterOptimization (HPO), since choice of hyperparameters has a significant impact onfinal model performance. Although there are many HPO algorithms, they eitherhave no theoretical guarantees or require strong assumptions. To this end, weintroduce BLiE -- a Lipschitz-bandit-based algorithm for HPO that only assumesLipschitz continuity of the objective function. BLiE exploits the landscape ofthe objective function to adaptively search over the hyperparameter space.Theoretically, we show that $(i)$ BLiE finds an $\epsilon$-optimalhyperparameter with $\mathcal{O} \left( \epsilon^{-(d_z + \beta)}\right)$ totalbudgets, where $d_z$ and $\beta$ are problem intrinsic; $(ii)$ BLiE is highlyparallelizable. Empirically, we demonstrate that BLiE outperforms thestate-of-the-art HPO algorithms on benchmark tasks. We also apply BLiE tosearch for noise schedule of diffusion models. Comparison with the defaultschedule shows that BLiE schedule greatly improves the sampling speed.

A Lipschitz Bandits Approach for Continuous Hyperparameter Optimization