Asymptotic Performance of Time-Varying Bayesian Optimization

TL;DR

This work analyzes the asymptotic regret performance of Time-Varying Bayesian Optimization (TVBO) under four popular stationary temporal kernel classes by exploiting the spectral properties of the spatio-temporal covariance operator. It shows that for broadband and band-limited kernels, any TVBO algorithm suffers linear cumulative regret, i.e., R_n ∈ Θ(n), while for almost-periodic and low-rank temporal kernels, the regret scales sublinearly, R_n ∈ o(n), under GP-UCB, with mutual information I(f_n, y_n) ∈ o(n). The analysis hinges on a spectral-density classification of k_T and establishes a Nyquist-type relationship for band-limited kernels, linking temporal sampling to achievable performance. These results unify regret analyses across kernel classes and identify practical no-regret conditions for TVBO, supported by numerical experiments.

Abstract

Time-Varying Bayesian Optimization (TVBO) is the go-to framework for optimizing a time-varying black-box objective function that may be noisy and expensive to evaluate, but its excellent empirical performance remains to be understood theoretically. Is it possible for the instantaneous regret of a TVBO algorithm to vanish asymptotically, and if so, when? We answer this question of great importance by providing upper bounds and algorithm-independent lower bounds for the cumulative regret of TVBO algorithms. In doing so, we provide important insights about the TVBO framework and derive sufficient conditions for a TVBO algorithm to have the no-regret property. To the best of our knowledge, our analysis is the first to cover all major classes of stationary kernel functions used in practice.

Figures (5)

• Figure 1: Spectra of $\bm K^{(n)}_S/n$ (left), $\bm K^{(n)}_T$ (center) and $\bm K^{(n)}$ (right) when $k_S$ and $k_T$ are RBF kernels and $n = 100$. The spectrum of each kernel matrix is plotted in blue and their $n$ largest products are plotted in orange. The eigenvalues in the spatial (left) and temporal (center) spectra involved in at least one of the $n$ largest products are colored in red. The spatial component $\bm x_i$ of an observation $\left(\bm x_i, t_i, y_i\right)$ is collected uniformly in $\mathcal{S} = [0, 1]^d$ while the temporal component is $t_i = i\Delta$.
• Figure 2: Spectrum of the temporal kernel matrix $\bm K_T^{(n)}$ (blue) and its approximation $\{S_T((j-n/2)/n\Delta)\}_{j \in [n]}$ provided by Proposition \ref{['prop:continuous_support_time_spectrum']} with the eigenvalues sorted (green) for different number of observations $n$, different sampling frequencies $\Delta$, with $k_T$ being an RBF kernel (top row) and a sinc2 kernel whose spectral density is supported on $[-\tau, \tau]$ (bottom row). The unsorted spectrum approximation is in orange.
• Figure 3: Spectrum of the temporal empirical kernel matrix $\bm K_T^{(n)}$ for a periodic kernel of period $r$, two commensurate sampling frequencies ($3/r$ and $6/r$) and two different numbers of observations.
• Figure 4: Mutual information $I(\bm f_n, \bm y_n)$ scaled by $n$ w.r.t. $n$ for four different temporal kernels, namely an RBF kernel (blue crosses), a sinc2 kernel (orange diamonds), a periodic kernel (green circles) and a low-rank kernel (red stars). The spatial components of observations are collected in $\mathcal{S} = [0, 1]^d$ while the temporal components follow Assumption \ref{['ass:sampling_freq']}. The results are averaged over 10 independent replications and standard error intervals are plotted as shaded areas around the solid lines.
• Figure 5: Comparison between a regular BO algorithm and the oracle built in this appendix. The temporal (resp., spatial) domain is represented by the x (resp., y)-axis. An arbitrary objective function $f$ is depicted in the background by a colored contour plot. The present running time is shown as a black vertical dashed line. (Left) At each iteration, a regular BO algorithm is allowed to observe a function value $f(\bm x, t)$ at a specific location in space-time $(\bm x, t) \in \mathcal{S} \times \mathcal{T}$ shown as red crosses. (Right) At each iteration, the oracle also queries a point $(\bm x, t)$ in space-time (shown with red crosses), but is allowed to observe the whole function $f(\cdot, t)$ on the spatial domain (shown with red vertical lines).

Theorems & Definitions (30)

• Proposition 3.1
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• Theorem 5.1
• Theorem 5.2
• proof
• proof
• Lemma C.1
• proof