qPOTS: Efficient batch multiobjective Bayesian optimization via Pareto optimal Thompson sampling

TL;DR

qPOTS tackles expensive multiobjective optimization by replacing costly acquisition-function optimization with Pareto-optimal Thompson sampling over GP posteriors for $K$ objectives, over $oldsymbol{x}\in\mathcal{X}\subset\mathbb{R}^d$. It samples posterior paths $Y_i(\cdot)$ and performs a cheap multiobjective optimization on each path (via NSGA-II) to obtain a Pareto set $X^*$, from which batch points are selected by a maximin distance rule, providing an automatic exploration–exploitation balance. The method extends to constrained problems by introducing constraint GPs and feasibility indicators, and scales with Nyström approximations to reduce covariance costs to $\mathcal{O}(m^3+Nm^2)$. Theoretical guarantees show asymptotic consistency of the GP posteriors and the converged Pareto frontier, and extensive experiments on synthetic benchmarks and real-world problems demonstrate superior sample efficiency and batch performance compared to state-of-the-art MOBO methods.

Abstract

Classical evolutionary approaches for multiobjective optimization are quite accurate but incur a lot of queries to the objectives; this can be prohibitive when objectives are expensive oracles. A sample-efficient approach to solving multiobjective optimization is via Gaussian process (GP) surrogates and Bayesian optimization (BO). Multiobjective Bayesian optimization (MOBO) involves the construction of an acquisition function which is optimized to acquire new observation candidates sequentially. This ``inner'' optimization can be hard due to various reasons: acquisition functions being nonconvex, nondifferentiable and/or unavailable in analytical form; batch sampling usually exacerbates these problems and the success of MOBO heavily relies on this inner optimization. This, ultimately, affects their sample efficiency. To overcome these challenges, we propose a Thompson sampling (TS) based approach ($q\texttt{POTS}$). Whereas TS chooses candidates according to the probability that they are optimal, $q\texttt{POTS}$ chooses candidates according to the probability that they are Pareto optimal. Instead of a hard acquisition function optimization, $q\texttt{POTS}~$ solves a cheap multiobjective optimization on the GP posteriors with evolutionary approaches. This way we get the best of both worlds: accuracy of evolutionary approaches and sample-efficiency of MOBO. New candidates are chosen on the posterior GP Pareto frontier according to a maximin distance criterion. $q\texttt{POTS}~$ is endowed with theoretical guarantees, a natural exploration-exploitation trade-off, and superior empirical performance.

Figures (11)

• Figure 1: Demonstration on the ZDT3 ($K=2$) test function that has disjoint Pareto frontiers. Navy are the true Pareto frontier and red are the predicted Pareto frontier. With $20$ seed points and only an additional $204$ acquisitions, in batches of $q=4$, notice that $q\texttt{POTS}$ is quickly able to resolve the true Pareto frontier while other methods in the state of the art struggle to come any close.
• Figure 2: Illustration of $q\texttt{POTS}$ in the constrained setting. Brown circles are GP training points, blue circles are $X^*$, black crosses are the true Pareto frontier/set, black line is the constraint boundary and blue square is the $q\texttt{POTS}$ acquisition. Left: output space, right: input space.
• Figure 3: Sequential acquisition. Hypervolume Vs. iterations for sequential ($q=1$) acquisition; plots show mean and $\pm 1$ standard deviation out of $10$ repetitions. $q\texttt{POTS}$ outperforms all competitors or is amongst the best. Bottom right shows constrained case on the OSY problem osyczka1995new. Additional experiments are provided in the supplementary material.
• Figure 4: Batch acquisition. Hypervolume Vs. iterations for batch ($q>1$) acquisition; plots show mean and $\pm 1$ standard deviation out of $10$ repetitions. $q\texttt{POTS}$ outperforms all competitors, but the benefit is more pronounced in the batch case. Additional experiments, including constrained problems, are shown in the supplementary material.
• Figure 5: Demonstration on the Branin-Currin ($d=2, K=2$) test function. Maroon: training points, blue circles: GP posterior samples, blue square: $q\texttt{POTS}$ acquisition, black: true Pareto frontier/set. The input space plots show how $q\texttt{POTS}$ explores and exploits the space (based on distance to maroon). The output space plots show how $q\texttt{POTS}$ asymptotically converges to the solution.

Theorems & Definitions (6)

• Lemma 3.3
• Theorem 3.4: Convergence with high probability
• Lemma 6.3
• proof : Proof
• Theorem 6.4: Convergence with high probability
• proof : Proof of Theorem 3.4