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Practical Global and Local Bounds in Gaussian Process Regression via Chaining

Junyi Liu, Stanley Kok

TL;DR

Addressing the lack of global uncertainty control in Gaussian Process Regression, the paper introduces a chaining-based framework to bound the expected global maximum without relying on input features or posterior estimates. It develops Talagrand-style chaining bounds on the kernel-induced metric, with kernel-specific refinements for RBF and Matérn that tighten constants and avoid relaxations. It also adds a locally adaptive uncertainty quantification method based on partition diameters to provide input-specific bounds without posterior variance scaling. Empirical results on synthetic and real datasets show tighter, more reliable bounds and favorable comparisons to state-of-the-art baselines, including improved Coverage Width-Based Criterion.

Abstract

Gaussian process regression (GPR) is a popular nonparametric Bayesian method that provides predictive uncertainty estimates and is widely used in safety-critical applications. While prior research has introduced various uncertainty bounds, most existing approaches require access to specific input features, and rely on posterior mean and variance estimates or the tuning of hyperparameters. These limitations hinder robustness and fail to capture the model's global behavior in expectation. To address these limitations, we propose a chaining-based framework for estimating upper and lower bounds on the expected extreme values over unseen data, without requiring access to specific input features. We provide kernel-specific refinements for commonly used kernels such as RBF and Matérn, in which our bounds are tighter than generic constructions. We further improve numerical tightness by avoiding analytical relaxations. In addition to global estimation, we also develop a novel method for local uncertainty quantification at specified inputs. This approach leverages chaining geometry through partition diameters, adapting to local structures without relying on posterior variance scaling. Our experimental results validate the theoretical findings and demonstrate that our method outperforms existing approaches on both synthetic and real-world datasets.

Practical Global and Local Bounds in Gaussian Process Regression via Chaining

TL;DR

Addressing the lack of global uncertainty control in Gaussian Process Regression, the paper introduces a chaining-based framework to bound the expected global maximum without relying on input features or posterior estimates. It develops Talagrand-style chaining bounds on the kernel-induced metric, with kernel-specific refinements for RBF and Matérn that tighten constants and avoid relaxations. It also adds a locally adaptive uncertainty quantification method based on partition diameters to provide input-specific bounds without posterior variance scaling. Empirical results on synthetic and real datasets show tighter, more reliable bounds and favorable comparisons to state-of-the-art baselines, including improved Coverage Width-Based Criterion.

Abstract

Gaussian process regression (GPR) is a popular nonparametric Bayesian method that provides predictive uncertainty estimates and is widely used in safety-critical applications. While prior research has introduced various uncertainty bounds, most existing approaches require access to specific input features, and rely on posterior mean and variance estimates or the tuning of hyperparameters. These limitations hinder robustness and fail to capture the model's global behavior in expectation. To address these limitations, we propose a chaining-based framework for estimating upper and lower bounds on the expected extreme values over unseen data, without requiring access to specific input features. We provide kernel-specific refinements for commonly used kernels such as RBF and Matérn, in which our bounds are tighter than generic constructions. We further improve numerical tightness by avoiding analytical relaxations. In addition to global estimation, we also develop a novel method for local uncertainty quantification at specified inputs. This approach leverages chaining geometry through partition diameters, adapting to local structures without relying on posterior variance scaling. Our experimental results validate the theoretical findings and demonstrate that our method outperforms existing approaches on both synthetic and real-world datasets.

Paper Structure

This paper contains 17 sections, 6 theorems, 90 equations, 3 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Gaussian Process Regression
  4. Chaining
  5. Related Work
  6. Upper and Lower Bounds
  7. Kernels
  8. Tighter Bounds
  9. Uncertainty Bounds
  10. Algorithm of Our Chaining Method
  11. Experiment
  12. Datasets
  13. Evaluation Metrics
  14. Baseline Settings
  15. Results
  16. ...and 2 more sections

Key Result

Theorem 1

talagrand2014upper (Eq 2.33) Let $T$ be an index set, $t_0 \in T$ be an initial index, $T_n \subseteq T$ for $n \ge 0$, and $T_0 = \{t_0\}$. For each $t \in T$, let $\pi_n(t) \in T_n$ for each $n \geq 0$, where each $\pi_n(t)$ represents a successive approximation of $t$, and let $\pi_n(t) = t$ for where $L$ is a universal constant (e.g., $L = \exp(9/2)$ satisfies the condition), $u \in \mathbb{R

Figures (3)

  • Figure 1: Comparison of our method with baselines for the test-point-specific bounds. The training set is in green, the test set in black, Lederer19 in orange, Fiedler21 in blue, Capone22 in purple, and our method in red.
  • Figure 2: An example of why chaining helps with supremum.
  • Figure 3: Comparison of our method with baselines for the test-point-specific bounds. The training set is in green, the test set in black, Lederer19 in orange, Fiedler21 in blue, Capone22 in purple, and our method in red.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • proof : Proof sketch.
  • Theorem 3
  • proof : Proof sketch.
  • Theorem 4
  • proof : Proof sketch.
  • Theorem 5
  • proof : Proof sketch.
  • Theorem 6
  • ...and 1 more