为非线性参数模型在贝叶斯优化中的理论分析提供统一框架。
Kernel-based guarantees for nonlinear parametric models in Bayesian optimization
- 基于参数空间的核函数构建再生核希尔伯特空间，支持非线性模型分析。
- 给出正则化凸损失下模型的置信区间，适用于自适应数据收集场景。
- 适用于随机正则化策略和复杂非线性模型，适合研究者与工程应用者。
现代贝叶斯优化与自适应采样方法越来越多依赖非线性参数模型，但针对自适应数据采集下这类模型的理论保证仍有限。现有分析多集中于高斯过程、核机器、线性模型或神经网络的线性近似，难以覆盖实际中使用的非线性模型。本文提出一种基于核函数的分析框架，用于研究在自适应数据上训练的正则化非线性参数模型。该方法通过在参数空间上定义核函数，诱导出对应模型类的再生核希尔伯特空间结构，从而为使用广泛正则化凸损失训练的模型提供置信区间。我们展示了这些边界如何支持非线性采集与代理模型的收敛性保证，包括通过最大化训练随机模型来选择采样点的随机正则化策略。结果为贝叶斯优化及相关自适应优化设置中非线性参数模型的统一分析提供了理论路径。
原文摘要 · Abstract (English)
Modern Bayesian optimization and adaptive sampling methods increasingly rely on nonlinear parametric models, yet theoretical guarantees for such models under adaptive data collection remain limited. Existing analyses largely focus on Gaussian processes, kernel machines, linear models, or linearized neural approximations, leaving a gap between theory and the nonlinear models used in practice. We develop a kernel based framework for analyzing regularized nonlinear parametric models trained on adaptively collected data. Our approach uses kernels over the parameter space to induce reproducing kernel Hilbert space structures over the corresponding model class, yielding confidence bounds for models trained with broad classes of regularized convex losses. We show how these bounds can support convergence guarantees for nonlinear acquisition and surrogate models, including randomized regularized policies that select points by maximizing a trained random model. These results provide a unified route to analyzing nonlinear parametric models in Bayesian optimization and related adaptive optimization settings.
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