Kernel-based guarantees for nonlinear parametric models in Bayesian optimization

2026-05-14 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, extended

Summary

This paper introduces a kernel-based framework for analyzing regularized nonlinear parametric models, such as deep neural networks, within Bayesian optimization (BO) and adaptive sampling settings. Current theoretical guarantees for these models are limited, often relying on Gaussian processes or linear approximations. The proposed framework uses kernels over the parameter space to induce Reproducing Kernel Hilbert Space (RKHS) structures over the model class, enabling the derivation of confidence bounds for models trained with various regularized convex losses. These bounds support convergence guarantees for nonlinear acquisition and surrogate models, including randomized regularized policies. Specifically, the framework provides a first regret guarantee for randomized nonlinear parametric models in a likelihood-free Bayesian optimization (LFBO) context, demonstrating sublinear cumulative regret for approximate Thompson sampling under finite-domain, realizable, and unique-optimum conditions. The analysis shows that pointwise approximation errors can be bounded by standard Gaussian Process quantities, even when the learned model is not a GP.

Key takeaway

For AI Scientists and Research Scientists developing or applying Bayesian optimization, this framework offers a principled route to analyze nonlinear parametric models. You can now derive confidence and regret bounds for models like deep neural networks in adaptive sampling, moving beyond traditional Gaussian Process assumptions. This enables the use of flexible, scalable parametric models with theoretical guarantees, particularly in likelihood-free BO settings, by understanding how random initialization and regularization contribute to exploration and stability.

Key insights

A kernel-based framework provides theoretical guarantees for nonlinear parametric models in adaptive optimization.

Principles

• RKHS structures enable control of nonlinear estimators.
• Random initialization and regularization induce exploration.
• GP-style bounds apply to non-GP models via kernel domination.

Method

The method embeds parametric models into an RKHS using parameter-space kernels, allowing regularization and control of prediction errors. It applies to randomized regularized policies where models are trained with random initializations and regularization penalties.

In practice

• Use RKHS for nonlinear model analysis in BO.
• Employ randomized initializations for exploration.
• Apply regularization to stabilize trained models.

Topics

• Bayesian Optimization
• Nonlinear Parametric Models
• Reproducing Kernel Hilbert Spaces
• Adaptive Data Collection
• Likelihood-Free Bayesian Optimization

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.