Poisson Process for Bayesian Optimization

TL;DR

PoPBO tackles the challenge that absolute objective values are often noisy or unavailable in black-box optimization by directly modeling global rankings with a non-homogeneous Poisson process. It learns an intensity function $\lambda_\xi(x;\theta)$ via an MLP to capture ranking structure and derives two acquisition functions, Rectified Lower Confidence Bound (R-LCB) and Expected Ranking Improvement (ERI), tailored for ranking-based surrogates. The method offers $O(N^2)$ computational cost and demonstrates robust, competitive performance on both synthetic benchmarks (e.g., $2$-D Branin, $6$-D Hartmann, $6$-D Rosenbrock) and real-world problems like HPO-Bench and NAS-Bench-201, including NAS. Overall, PoPBO provides a practical, noise-robust alternative to value-based BO with strong transferability across domains and fidelities.

Abstract

BayesianOptimization(BO) is a sample-efficient black-box optimizer, and extensive methods have been proposed to build the absolute function response of the black-box function through a probabilistic surrogate model, including Tree-structured Parzen Estimator (TPE), random forest (SMAC), and Gaussian process (GP). However, few methods have been explored to estimate the relative rankings of candidates, which can be more robust to noise and have better practicality than absolute function responses, especially when the function responses are intractable but preferences can be acquired. To this end, we propose a novel ranking-based surrogate model based on the Poisson process and introduce an efficient BO framework, namely Poisson Process Bayesian Optimization (PoPBO). Two tailored acquisition functions are further derived from classic LCB and EI to accommodate it. Compared to the classic GP-BO method, our PoPBO has lower computation costs and better robustness to noise, which is verified by abundant experiments. The results on both simulated and real-world benchmarks, including hyperparameter optimization (HPO) and neural architecture search (NAS), show the effectiveness of PoPBO.

Figures (11)

• Figure 1: We compare the sensitivity of additive Gaussian noise between GP (value-based) response surface and PoPBO (ranking-based response surface) on the Forrester function. Based on the function value, the solid black line (oracle) indicates actual rankings over 100 points evenly spaced from 0 to 0.8. We draw lines between the 100 predictions by linear interpolation for a clear illustration. The dashed lines indicate predicted rankings over the 100 points by (a) Gaussian process and (b) Poisson process on observations with varying degrees of noise whose standard deviation $\sigma$ ranges from $0$ to $0.45$. Each response surface is trained on the same 15 queries. Note that GP performs worse as the standard deviation of noise increases. In contrast, PP performs consistently well due to its great robustness against noise.
• Figure 2: Time cost of GP-BO, PPBO DBLP:ppbo_conf/icml/MikkolaTJRK20 and PoPBO. All the methods are applied to optimize 6-d Hartmann function. The units are wall-clock times.
• Figure 3: Performance of black-box optimization methods on three simulation functions. Y-axis is the residual of the optimum function value and the incumbent. We run each method ten times and plot the average performance and standard error as the line and shadow.
• Figure 4: Minimum regret comparison with random search and various Bayesian optimization methods on tabular datasets in HPO-Bench. Y-axis indicates the residual between the optimum function value and the incumbent. We run each method ten times and plot the average performance and standard error of the incumbent as the line and shadow. Our PoPBO can quickly discover good samples and achieves the best performance (lowest regret).
• Figure 5: Ablation study on hyperparameter $q$, which controls the exploitation-exploration trade-off. We test on the Rosenbrock simulation function and show the effect of $q$ on (a) R-LCB and (b) ERI. For each setting, we run ten times and plot the average performance of the incumbent.