Bayesian Optimization with a Fin

2020-03-19 本文已影响0人馒头and花卷

Lam R, Willcox K, Wolpert D H, et al. Bayesian Optimization with a Finite Budget: An Approximate Dynamic Programming Approach[C]. neural information processing systems, 2016: 883-891.

@article{lam2016bayesian,
title={Bayesian Optimization with a Finite Budget: An Approximate Dynamic Programming Approach},
author={Lam, Remi and Willcox, Karen and Wolpert, David H},
pages={883--891},
year={2016}}

概

贝叶斯优化中的多步优化策略. 像经典的EI方法, 就是只考虑一步, 即希望找到
$r(\mathcal{S}_k, x_{k+1},f_{k+1})=\max \{0, f_{min}^{\mathcal{S}_k}-f_{k+1}\}$
的期望收益最大化的点 $x_{k+1}$ 为下一个评估点.

上式中的 $f_{min}^{\mathcal{S}_k}$ 是指目标函数在集合 $\mathcal{S}_k$ 上的最小值.

主要内容

考虑如下动态规划, 第k步的
状态: $\mathcal{S}_k$ , 即观测到的点;
控制: $u_k$ , 且 $u_k(\mathcal{S}_k)=x_{k+1}$
扰动: $w_k:=f_{k+1} \sim p(f(x_{k+1})|\mathcal{S}_k)$ ;

设状态转移为:
$\mathcal{S}_{k+1} = \mathcal{F}_k (\mathcal{S}_{k}, x_{k+1}, f_{k+1}) = \mathcal{S}_{k}\cup \{(x_{k+1}, f_{k+1})\}.$

收益(效用函数):
$U_k(x_{k+1}; \mathcal{S} _k) = \mathbb{E}_{w_k}[r_k(\mathcal{S}_k, x_{k+1}, f_{k+1})+J_{k+1}(\mathcal{F}_k (\mathcal{S}_{k}, x_{k+1}, f_{k+1}))], \\ J_k(x_{k+1}) = \max_{x_{k+1}} U_k,\\ J_N=r_N(x_{N+1}).$