Event-Triggered Time-Varying Bayesian Optimization

TL;DR

The paper tackles sequential optimization of a time-varying objective with an unknown rate of change $\varepsilon$ by introducing ET-GP-UCB, an event-triggered Bayesian optimization algorithm. ET-GP-UCB uses probabilistic uniform error bounds to detect model mismatch and resets its dataset within an admissible window to adapt to realized changes, delivering regret bounds that depend on the unknown variation via functions like $\phi_T(\varepsilon,\bar{N})$ without requiring exact $\varepsilon$. Theoretical results extend prior TVBO regret analyses to adaptive resets and are complemented by extensive empirical evaluations on synthetic benchmarks and real-world data (e.g., temperature sensing, policy search), showing superior performance and reduced hyperparameter tuning. The work also provides practical extensions, including online hyperparameter tuning strategies and mechanisms to retain past information, enhancing data efficiency in higher dimensions and more complex settings.

Abstract

We consider the problem of sequentially optimizing a time-varying objective function using time-varying Bayesian optimization (TVBO). Current approaches to TVBO require prior knowledge of a constant rate of change to cope with stale data arising from time variations. However, in practice, the rate of change is usually unknown. We propose an event-triggered algorithm, ET-GP-UCB, that treats the optimization problem as static until it detects changes in the objective function and then resets the dataset. This allows the algorithm to adapt online to realized temporal changes without the need for exact prior knowledge. The event trigger is based on probabilistic uniform error bounds used in Gaussian process regression. We derive regret bounds for adaptive resets without exact prior knowledge of the temporal changes and show in numerical experiments that ET-GP-UCB outperforms competing GP-UCB algorithms on both synthetic and real-world data. The results demonstrate that ET-GP-UCB is readily applicable without extensive hyperparameter tuning.

Figures (16)

• Figure 1: Illustration of our proposed concept. After the standard steps of optimizing the acquisition function and obtaining an observation, an event trigger decides whether to reset the data set ($\gamma_{\mathrm{reset}} = 1$) if a change is detected (see red cross), or augment it with the observed data point in the usual way ($\gamma_{\mathrm{reset}} = 0$). This event trigger allows our algorithm ET-GP-UCB to be adaptive to changes in the objective function without relying on exact prior knowledge of the rate of change.
• Figure 2: Left: Influence of the window size on the empirical PDF of reset times. For small window sizes, the probability mass in the tails is cut off through $\underaccent{\bar{}}{N}$ and $\bar{N}$. As the window size $W$ increases, hence the flexibility of our event trigger increases, the histogram resembles an inverse-gamma-like distribution. Right: Influence of the window size on the empirical regret for different $\varepsilon$. As the window size increases, the regret of ET-GP-UCB (markers) decreases always staying below R-GP-UCB (dash-dotted lines). Increasing the flexibility of our trigger empirically reduces regret. Always resetting at $\underaccent{\bar{}}{N}$ (dashed line) shows increased empirical regret. This indicates that Corollary \ref{['coro:scaling']} is a relatively loose bound for our trigger.
• Figure 3: Real-world examples. Subfigure (a) shows the temperature data from the test days used in Bogunovic2016. In (b), the performance of the algorithms on these two days (top) and the resets of R-GP-UCB and ET-GP-UCB (bottom) are displayed. Subfigure (c) shows the performance on other test days. ET-GP-UCB displays superior performance improvement compared to all other algorithms. For these experiments on real-world data, we use the same parameter for the event trigger as in all previous experiments whereas TV-GP-UCB and R-GP-UCB relied on estimating $\varepsilon$. Lastly, in subfigure (d), ET-GP-UCB also outperforms all competing baselines on a policy search benchmark. For all methods, the median performance and the interquartile ($25\%$ and $75\%$) are displayed.
• Figure 4: Impact of Reset Frequency vs. Timing.
• Figure 5: Left: Performance on an eight-dimensional within-model objective function. Retaining a portion of past data using Algorithm \ref{['alg:backtracking']} leads to a significant performance boost, nearly matching the results of TV-GP-UCB, even without prior knowledge of $\varepsilon$. Right: Final performance for different dimensions ($d \in \{2, 4, 6, 8\}$) with a fixed $\varepsilon$. As dimensionality increases, the benefit of retaining data becomes more pronounced.

Theorems & Definitions (16)

• Remark 1
• Definition 1
• Theorem 1
• Corollary 1
• Definition 2: Event trigger in
• Lemma 1: Srinivas2010
• Remark 2
• Lemma 2
• Lemma 3
• Lemma 4: Bogunovic2016