No-Regret Gaussian Process Optimization of Time-Varying Functions

2026-07-08 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

The paper introduces W-SparQ-GP-UCB, a novel Gaussian Process bandit algorithm designed for sequential optimization of black-box, time-varying functions from noisy evaluations. Traditional GP bandit algorithms fail to guarantee no-regret in non-stationary settings without strong assumptions. W-SparQ-GP-UCB addresses this by capturing time variations through uncertainty injection (UI), enabling heteroscedastic GP regression. It achieves no-regret with only a vanishing number of additional queries per iteration, relaxing the strict bandit setting. The method establishes a lower bound on required additional queries, proving its efficiency. The analysis links the function's time-variation degree, parameterized by alpha, to achievable regret rates, identifying slow (alpha < 1/3), moderate (1/3 <= alpha <= 1), and fast (alpha > 1) variation regimes.

Key takeaway

For AI Scientists and Research Scientists optimizing dynamic black-box systems, this work demonstrates that achieving no-regret with time-varying functions necessitates a strategic relaxation of the strict bandit setting. You should evaluate your system's temporal variability using the alpha parameter; if alpha >= 1/3, standard GP-UCB will incur growing regret. Implement W-SparQ-GP-UCB to utilize sparse, windowed additional queries, ensuring sublinear regret while minimizing costly expert interactions.

Key insights

No-regret optimization of time-varying functions requires a vanishing number of additional queries, especially for rapid changes.

Principles

• No-regret is unattainable in strict bandit settings for persistently time-varying functions.
• Temporal variations can be modeled via uncertainty injection, adapting past observations.
• The parameter alpha dictates three distinct regimes of function variability and regret.

Method

W-SparQ-GP-UCB uses uncertainty injection for heteroscedastic GP regression, selects sparse, informative locations via a Determinantal Point Process (DPP), and obtains expert feedback to refresh posteriors, reducing expert calls through windowing.

In practice

• Apply W-SparQ-GP-UCB for optimizing real-world systems like recommender systems or sensor networks.
• Consider the alpha parameter to determine if additional queries are necessary for no-regret.
• Utilize sparse inference and windowing to minimize costly expert feedback in dynamic environments.

Topics

• Gaussian Processes
• Time-varying Optimization
• Bandit Algorithms
• No-Regret Learning
• Uncertainty Injection
• Determinantal Point Process

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.