We propose a practical Bayesian optimization method over sets, to minimize a black-box function that takes a set as a single input. Because set inputs are permutation-invariant, traditional Gaussian process-based Bayesian optimization strategies which assume vector inputs can fall short. To address this, we develop a Bayesian optimization method with set kernel that is used to build surrogate functions. This kernel accumulates similarity over set elements to enforce permutation-invariance, but this comes at a greater computational cost. To reduce this burden, we propose two key components: (i) a more efficient approximate set kernel which is still positive-definite and is an unbiased estimator of the true set kernel with upper-bounded variance in terms of the number of subsamples, (ii) a constrained acquisition function optimization over sets, which uses symmetry of the feasible region that defines a set input. Finally, we present several numerical experiments which demonstrate that our method outperforms other methods.

我们提出了一种针对集合的实用贝叶斯优化方法，以最小化将集合作为单个输入的黑盒函数。由于集合输入是不变排列的，因此假设矢量输入的传统基于高斯过程的贝叶斯优化策略是不可行的。为了解决这个问题，我们开发了具有集合核的贝叶斯优化方法用于构建代理功能。该内核会在集合元素上积累相似性以实施置换不变性，但这会带来更大的计算成本。为了减轻这种负担，我们提出了两个关键组成部分：（i）一个更有效的近似集核，该核仍然是正定的，并且是真实集核的无偏估计量，在子样本数量上，方差具有上限，（ ii）对集合的约束获取函数优化，它使用定义集合输入的可行区域的对称性。最后，我们提出了几个数值实验，证明了我们的方法优于其他方法。