We introduce a surrogate-based black-box optimization method, termed
Polynomial-model-based optimization (PMBO). The algorithm alternates polynomial
approximation with Bayesian optimization steps, using Gaussian processes to
model the error between the objective and its polynomial fit. We describe the
algorithmic design of PMBO and compare the results of the performance of PMBO
with several optimization methods for a set of analytic test functions.
The results show that PMBO outperforms the classic Bayesian optimization and
is robust with respect to the choice of its correlation function family and its
hyper-parameter setting, which, on the contrary, need to be carefully tuned in
classic Bayesian optimization. Remarkably, PMBO performs comparably with
state-of-the-art evolutionary algorithms such as the Covariance Matrix
Adaptation -- Evolution Strategy (CMA-ES). This finding suggests that PMBO
emerges as the pivotal choice among surrogate-based optimization methods when
addressing low-dimensional optimization problems. Hereby, the simple nature of
polynomials opens the opportunity for interpretation and analysis of the
inferred surrogate model, providing a macroscopic perspective on the landscape
of the objective function.

采用多项式模型为基础的优化方法 (PMBO) 在低维优化问题中表现出色，与经典贝叶斯优化和进化算法相比具有更好的鲁棒性和解释性。

PMBO: 通过多元多项式替代物强化黑箱优化

PMBO: Enhancing Black-Box Optimization through Multivariate Polynomial  Surrogates

For a wide range of applications the structure of systems like Neural
Networks or complex simulations, is unknown and approximation is costly or even
impossible. Black-box optimization seeks to find optimal (hyper-) parameters
for these systems such that a pre-defined objective function is minimized.
Polynomial-Model-Based Optimization (PMBO) is a novel blackbox optimizer that
finds the minimum by fitting a polynomial surrogate to the objective function.
Motivated by Bayesian optimization the model is iteratively updated according
to the acquisition function Expected Improvement, thus balancing the
exploitation and exploration rate and providing an uncertainty estimate of the
model. PMBO is benchmarked against other state-of-the-art algorithms for a
given set of artificial, analytical functions. PMBO competes successfully with
those algorithms and even outperforms all of them in some cases. As the results
suggest, we believe PMBO is the pivotal choice for solving blackbox
optimization tasks occurring in a wide range of disciplines.

通过拟合多项式替代模型到目标函数，通过迭代更新模型并利用预期改进的探索和利用平衡，提供模型的不确定性估计的 PMBO 是一种成功竞争，并在某些情况下甚至超越其他算法的黑盒优化方法，可广泛应用于各个领域。