We study the problem of selecting $k$ experiments from a larger candidate pool, where the goal is to maximize mutual information (MI) between the selected subset and the underlying parameters. Finding the exact solution is to this combinatorial optimization problem is computationally costly, not only due to the complexity of the combinatorial search but also the difficulty of evaluating MI in nonlinear/non-Gaussian settings. We propose greedy approaches based on new computationally inexpensive lower bounds for MI, constructed via log-Sobolev inequalities. We demonstrate that our method outperforms random selection strategies, Gaussian approximations, and nested Monte Carlo (NMC) estimators of MI in various settings, including optimal design for nonlinear models with non-additive noise.

翻译：本文研究从较大候选池中选取$k$个实验的问题，目标是最优参数与被选子集之间的互信息。对该组合优化问题而言，求解精确解的计算成本很高，这不仅源于组合搜索的复杂性，还源于在非线性/非高斯设置中评估互信息的困难性。我们提出了基于新计算成本较低互信息下界的贪婪方法，该下界通过对数Sobolev不等式构建。我们证明，在多种设置（包括具有非加性噪声的非线性模型的最优设计）中，我们的方法优于随机选择策略、高斯近似以及互信息的嵌套蒙特卡洛估计器。