基于核的方法在贝叶斯优化中非线性参数模型的理论保证
Kernel-based guarantees for nonlinear parametric models in Bayesian optimization
AI总结：
本文提出一种基于核的方法，用于分析在自适应数据收集下训练的正则化非线性参数模型，提供收敛保证。
摘要：
现代贝叶斯优化和自适应采样方法越来越多地依赖于非线性参数模型，但此类模型在自适应数据收集下的理论保证仍然有限。现有分析主要集中在高斯过程、核机、线性模型或线性化神经近似上，存在理论与实践中非线性模型之间的差距。我们开发了一种基于核的框架，用于分析在自适应数据上训练的正则化非线性参数模型。我们的方法利用参数空间上的核诱导相应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类训练的模型提供置信界。我们展示了这些界如何支持非线性获取和替代模型的收敛保证，包括通过最大化训练随机模型选择点的随机正则化策略。这些结果为贝叶斯优化及相关自适应优化设置中的非线性参数模型分析提供了一条统一的路线。
英文摘要：
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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