[Paper] Consequences of Kernel Regularity for Bandit Optimization

Source: arXiv - 2512.05957v1

Overview

This paper digs into why the shape of a kernel matters when you use it for bandit‑style optimization of black‑box functions. By linking the spectral decay of popular kernels (Matérn, SE, Rational‑Quadratic, etc.) to both global Gaussian‑process (GP) methods and local polynomial approximations, the authors show that seemingly different algorithms are actually two sides of the same coin. The result is a unified way to predict regret (the cost of not picking the optimum) across a wide range of kernels and to design hybrid methods that work well in practice.

Key Contributions

• Spectral‑based unification: Demonstrates that the Fourier decay rate of an isotropic kernel simultaneously governs (i) the maximum information gain for GP‑UCB‑style bandits and (ii) the Hölder/Besov smoothness needed for local polynomial estimators.
• Explicit regret formulas: Derives closed‑form, asymptotically tight regret bounds for six major kernel families, including several (e.g., γ‑exponential, piecewise‑polynomial) that lacked prior bandit analyses.
• Hybrid algorithm analysis (LP‑GP‑UCB): Provides the first theoretical guarantees for a method that blends a global GP surrogate with locally fitted polynomials, showing order‑optimal regret for multiple kernels.
• Practical embedding insights: Shows how to translate a kernel’s spectral decay into concrete smoothness parameters (Hölder exponents), enabling developers to pick kernels based on desired locality vs. globality trade‑offs.
• Improved bounds for existing methods: Tightens previous regret results for standard GP‑UCB and locally adaptive algorithms by exploiting the unified spectral perspective.

Methodology

1. Spectral Characterization:

   • For each isotropic kernel (k(r)) the authors compute its Fourier transform (\hat{k}(\omega)) and derive the asymptotic decay (\hat{k}(\omega) \asymp |\omega|^{-\beta}).
   • The exponent (\beta) directly informs two mathematical objects:
     • The maximum information gain (\gamma_T) (a GP‑UCB quantity) scales as (\tilde{O}(T^{d/\beta})).
     • The Hölder smoothness of functions in the associated RKHS, which determines how well local polynomial fits approximate the function.

2. Regret Analysis:

   • Using known GP‑UCB regret bounds (R_T = O(\sqrt{T\gamma_T})), they substitute the kernel‑specific (\gamma_T) to obtain explicit regret rates.
   • For local methods, they employ Besov space embeddings to bound the error of polynomial estimators, then translate those bounds into regret via a standard “optimism‑in‑the‑face‑of‑uncertainty” argument.

3. Hybrid LP‑GP‑UCB:

   • The algorithm maintains a global GP posterior for exploration and, in parallel, fits a low‑degree polynomial in a neighborhood around the current best point.
   • The decision rule picks the point with the highest combined upper confidence bound (GP‑UCB term + polynomial‑based term).
   • The analysis shows that the combined bound inherits the best decay exponent among the kernels considered, yielding order‑optimal regret across them.

4. Unified Proof Skeleton:

   • The authors construct a “spectral decay → smoothness → regret” pipeline that works for any isotropic kernel satisfying mild regularity (e.g., integrable Fourier transform).

Results & Findings

• LP‑GP‑UCB matches the best of the above bounds for each kernel, achieving order‑optimal regret without knowing the kernel’s smoothness a priori.
• The analysis also reveals that for kernels with polynomial decay (e.g., Matérn with low (\nu)), purely global GP‑UCB can be sub‑optimal compared to a hybrid that leverages local smoothness.

Practical Implications

• Kernel selection becomes quantitative: Developers can now look at a kernel’s spectral decay (often listed in libraries) to estimate the regret they can expect in bandit‑type hyperparameter tuning, Bayesian optimization, or active learning.
• Hybrid algorithms are ready for production: LP‑GP‑UCB’s theoretical guarantees translate into a simple recipe—run a standard GP‑UCB loop and, every few iterations, fit a low‑degree polynomial around the current best point. This adds negligible overhead but can dramatically improve performance on moderately smooth functions.
• Better default settings for AutoML tools: AutoML frameworks that currently default to SE kernels can consider Matérn or Rational‑Quadratic kernels with tuned smoothness parameters to balance exploration speed and computational cost.
• Guidance for high‑dimensional problems: The regret formulas expose the curse of dimensionality via the (d/\beta) term, encouraging practitioners to either (i) reduce dimensionality (e.g., via embeddings) or (ii) pick kernels with faster decay (SE, γ‑Exponential) when feasible.
• Interpretability of uncertainty estimates: Understanding that the GP posterior’s information gain is tied to spectral decay helps engineers debug overly optimistic confidence bounds—if the kernel’s decay is slow, the GP may be under‑confident in unexplored regions.

Limitations & Future Work

• Isotropic assumption: The analysis hinges on kernels that are radially symmetric. Real‑world data often benefit from anisotropic or learned kernels, which may exhibit mixed spectral decay. Extending the framework to those cases remains open.
• Asymptotic focus: Regret bounds are derived for large (T); constant factors (e.g., kernel hyper‑parameter estimation error) are not captured, which can dominate early iterations in practice.
• Computational scaling: While LP‑GP‑UCB adds only a cheap local polynomial fit, the underlying GP still suffers from cubic scaling. Integrating sparse GP approximations into the unified analysis is a natural next step.
• Empirical validation: The paper is primarily theoretical. Benchmarking LP‑GP‑UCB against state‑of‑the‑art Bayesian optimization libraries (e.g., BoTorch, Ax) on real hyperparameter tuning tasks would solidify its practical impact.

Bottom line: By exposing the hidden link between a kernel’s Fourier decay and both global and local bandit strategies, this work equips developers with a principled way to choose kernels, anticipate performance, and deploy hybrid optimizers that are robust across a spectrum of smoothness scenarios.

Authors

• Madison Lee
• Tara Javidi

Paper Information

• arXiv ID: 2512.05957v1
• Categories: stat.ML, cs.LG
• Published: December 5, 2025
• PDF: Download PDF