Low Regret Binary Sampling Method for Efficient Global Optimization of Univariate Functions
This work addresses computational efficiency in optimization for researchers and practitioners, though it is incremental as it builds on existing methods like Piyavskii-Shubert with similar regret results.
The authors tackled global optimization of univariate functions by proposing a computationally efficient binary sampling algorithm that reduces the need for solving additional optimization problems, achieving at most L log(3T) regret for L-Lipschitz continuous functions and 2.25H regret for H-Lipschitz smooth functions.
In this work, we propose a computationally efficient algorithm for the problem of global optimization in univariate loss functions. For the performance evaluation, we study the cumulative regret of the algorithm instead of the simple regret between our best query and the optimal value of the objective function. Although our approach has similar regret results with the traditional lower-bounding algorithms such as the Piyavskii-Shubert method for the Lipschitz continuous or Lipschitz smooth functions, it has a major computational cost advantage. In Piyavskii-Shubert method, for certain types of functions, the query points may be hard to determine (as they are solutions to additional optimization problems). However, this issue is circumvented in our binary sampling approach, where the sampling set is predetermined irrespective of the function characteristics. For a search space of $[0,1]$, our approach has at most $L\log (3T)$ and $2.25H$ regret for $L$-Lipschitz continuous and $H$-Lipschitz smooth functions respectively. We also analytically extend our results for a broader class of functions that covers more complex regularity conditions.