No-Regret Gaussian Process Optimization of Time-Varying Functions

This paper studies the regret analysis for parallel Gaussian process (GP) bandit optimization. The known regret upper bounds for the widely used GP batched upper confidence bound and GP batched Thompson sampling (GP-BTS) suffer from a multiplicative factor with respect to the batch size $Q$. To avoid this degradation, existing analyses require a polynomial number of uncertainty sampling (US) for $Q$ at the beginning of optimization. However, this initial US phase is often ineffective in practice. This paper shows that the regret upper bound without the multiplicative factor on $Q$ can be achieved without the initial US phase, using GP-BTS as an example. Furthermore, we show much better regret upper bounds in the noiseless setting than in the noisy setting, as in the sequential GP bandit setting.

Bayesian optimization (BO) is a widely used framework for optimizing expensive black-box functions, commonly based on Gaussian process (GP) surrogate models. Its effectiveness relies on uncertainty quantification that is both sharp (informative) and well-calibrated along the BO trajectory. In practice, GP kernel hyperparameters are unknown and are refit online from sequentially collected (non-i.i.d.) data, which can yield miscalibrated or overly conservative uncertainty and lies outside the fixed-kernel assumptions of standard BO regret theory. We propose Online Sharp-Calibrated Bayesian Optimization (OSCBO), a BO algorithm that adaptively balances GP sharpness and calibration by casting hyperparameter selection as a constrained online-learning problem. We also show that OSCBO preserves sublinear regret bounds by leveraging the theoretical guarantees of the underlying online learning algorithm. Empirically, OSCBO performs competitively across synthetic and real-world benchmarks, ranking among the strongest methods in final simple regret while maintaining robust cumulative-regret behavior.

Many deployed systems expose black-box objectives whose minimizing configuration shifts with an externally observed context. When contexts revisit a small set of latent regimes, an optimizer that discards history pays repeated adaptation cost; when each step must remain inexpensive, full Gaussian-process (GP) refits at high observation counts are difficult to sustain. We cast online tuning as context-conditioned regret minimization and present RASP-Tuner, which instantiates a decomposition motivated by first principles: (i) identify a regime proxy by retrieving similar past contexts; (ii) predict short-horizon loss with a mixture-of-experts surrogate whose input concatenates parameters, context, and a retrieved soft prompt; (iii) adapt chiefly in a low-dimensional prompt subspace, invoking full surrogate updates only when scalarized error or disagreement spikes. A RealErrorComposer maps heterogeneous streaming metrics to [0,1] via EMA-stabilized logistic scores, supplying a single differentiable training target. On nine synthetic non-stationary benchmarks, an adversarial-context sanity check, and three tabular real-world streams (Section on real-world experiments), RASP-Tuner improves or matches cumulative regret relative to our GP-UCB and CMA-ES implementations on seven of nine synthetic tasks under paired tests at horizon T=100, while recording 8-12 times lower wall-clock per step than sliding-window GP-UCB on identical hardware. Idealized analysis in a cluster-separated, strongly convex regime model (RA-GD) supplies sufficient conditions for bounded dynamic regret; the deployed pipeline violates several of these premises, and we articulate which gaps remain open.