Beyond Grids: Multi-objective Bayesian Optimization With Adaptive Discretization
Authors: Andi Nika, Sepehr Elahi, Cagin Ararat, Cem Tekin
TMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | We also experimentally show that our algorithm outperforms other Pareto set identification methods. Furthermore, we also show via simulations on multi-objective functions that Adaptive ϵ-PAL significantly improves over its competitors in terms of accuracy. This section empirically evaluates Adaptive ϵ-PAL, comparing its performance and efficiency against other multi-objective Bayesian optimization (MOBO) methods. |
| Researcher Affiliation | Academia | Andi Nika EMAIL Max Planck Institute for Software Systems Germany Sepehr Elahi EMAIL Department of Computer and Communication Sciences EPFL Switzerland Çağın Ararat EMAIL Department of Industrial Engineering Bilkent University Turkey Cem Tekin EMAIL Department of Electrical and Electronics Engineering Bilkent University Turkey |
| Pseudocode | Yes | The pseudocode is given in Algorithm 1. Algorithm 1 Adaptive ϵ-PAL |
| Open Source Code | No | We used the implementations for PESMO and Par EGO from the Spearmint Bayesian optimization library4, for USe MO from the authors open-source code5, and for q NEHVI from the Bo Torch library.6 For completeness, we also attempted to benchmark against the original ϵ-PAL algorithm (Zuluaga et al., 2016). However, we encountered difficulties achieving termination when running both the authors publicly available code and our own implementation based on the published pseudocode. |
| Open Datasets | No | We simulate a problem with a 1-dimensional input space X = [0, 1] and a 2-dimensional objective space (m = 2). The objective function f is sampled from a GP with a zero mean function and independent squared exponential kernels for each objective: k1(x, x ) = 0.5 exp( (x x )2 2 0.12 ) and k2(x, x ) = 0.1 exp( (x x )2 2 0.062 ). Observation noise is Gaussian with σ = 0.01 (variance σ2 = 10 4). |
| Dataset Splits | No | To compute the metrics, we approximate the true Pareto front by sampling 10, 000 points uniformly from the input space, evaluating the objective function at these points, and identifying the Pareto optimal designs among them. ... This aligns well with the theoretical expectation of producing a valid ϵ-accurate Pareto set with high probability. The slight deviation from 100% can be attributed to the probabilistic nature of the guarantees (δ = 0.05) and the approximations involved in discretizing the true Pareto front for metric calculation. |
| Hardware Specification | No | The paper does not explicitly describe the hardware used to run its experiments. |
| Software Dependencies | No | We used the implementations for PESMO and Par EGO from the Spearmint Bayesian optimization library4, for USe MO from the authors open-source code5, and for q NEHVI from the Bo Torch library.6 ... The acquisition function optimization starts from the best point on a grid of size 1000, followed by L-BFGS optimization. ... The acquisition function is optimized using multi-start L-BFGS-B, consistent with the other baselines. |
| Experiment Setup | Yes | Adaptive ϵ-PAL. We run Adaptive ϵ-PAL with a target accuracy ϵ = (0.05, 0.05) and confidence δ = 0.05. The tree parameters were set to N = 2, ρ = 1/2, and v1 = v2 = 1. To investigate the impact of the maximum tree depth, we tested three settings for hmax. The first setting used the theoretically derived value hmax = 24, calculated using Lemma 4 based on the target ϵ. The other two settings used practical, reduced depths of hmax = 10 and hmax = 9 to evaluate the trade-off between computational cost and performance. When using these practical depth limits, refinement beyond hmax was prevented by setting Vh = 0 for all h hmax. PESMO and Par EGO. We run PESMO and Par EGO for 100 iterations (exceeding Adaptive ϵ-PAL s evaluations). We use the same δ = 0.05 where applicable. The acquisition function optimization starts from the best point on a grid of size 1000, followed by L-BFGS optimization. USe MO. We run USe MO for 100 iterations. Expected Improvement (EI) is used as the acquisition function, aggregated using the Tchebycheff scalarization as recommended by the authors. q NEHVI. We run the Noisy Expected Hypervolume Improvement (q NEHVI) algorithm for 100 iterations. We use the sequential setting (batch size q = 1) and the official implementation in Bo Torch. The acquisition function is optimized using multi-start L-BFGS-B, consistent with the other baselines. |