Optuna Constrained Tree-Structured Parzen Estimator Is a Joint Density Generalization of c-TPE

Expensive constrained optimization problems in real-world industry design often involve constraint thresholds that are difficult to determine in advance. Engineers may need to adjust constraint thresholds to explore different feasibility-performance trade-offs, requiring solutions under a wide range of threshold settings. However, existing constrained Bayesian optimization methods treat each threshold configuration independently, leading to repeated optimization and failing to exploit the shared relationship among continuously varying thresholds. To address this challenge, we propose constraint-bound agnostic Bayesian optimization (CBA-BO), a learning-based framework that learns a parametric constraint model mapping thresholds to optimal solutions. Once learned, CBA-BO directly predicts solutions for arbitrary unseen threshold configurations without additional optimization, with a one-step Bayesian optimization refinement further improving solution quality. Experiments on benchmark and engineering problems demonstrate that CBA-BO learns a transferable threshold-solution mapping, enabling efficient prediction and optimization for arbitrary threshold queries. An intent-guided constraint-bound recommendation mechanism is further developed to improve objective performance while satisfying user-specified constraint preferences.

Hyperparameter optimization (HPO) for Random Forest faces a specific difficulty in tuning the number of trees: the predictive score typically improves monotonically with ensemble size, so standard methods such as Tree-structured Parzen Estimator (TPE) and Hyperband require a predefined search range and often drive the estimate toward its right boundary. Early-stopping strategies avoid fixing such a range, but can be sensitive to score noise and prone to premature stopping. To address this, we propose an integrated triplet-based plateau-search algorithm that removes the number of trees from the direct TPE search space and still exploits information accumulated across HPO trials. The method adaptively tracks a near-minimal sufficient ensemble size by monitoring relative changes in the out-of-bag (OOB) score across a triplet of forest sizes and shifting this triplet accordingly. This yields an automated and user-interpretable procedure based on a tolerance parameter. We also provide a theoretical analysis: we relate the proposed relative OOB-score criterion to the gap between the current and limiting scores, and derive an asymptotic variance estimate for the corresponding OOB-based absolute relative difference. Experiments show that the selected number of trees can differ substantially from the common heuristic: for most classical benchmark datasets it is smaller, whereas for some high-dimensional bioinformatics datasets, such as Arcene and Dorothea, it is larger. The source code and reproducible experiments are available at https://github.com/lange-am/rf_plateau_hpo.

Bayesian conformal optimisation methods often use the same held-out data both to search for efficient prediction sets and to certify coverage or risk. This coupling is natural for high-probability risk-control guarantees, but it is not necessary when the target is standard finite-sample marginal conformal coverage. We propose Decoupled Conformal Optimisation (DCO), a train-tune-calibrate design principle that uses an independent tuning split for efficiency-oriented structural selection and a fresh calibration split for the final conformal quantile. Conditional on the tuned structure, standard split-conformal exchangeability yields finite-sample marginal coverage for any candidate class, without a confidence parameter or multiple-testing correction. DCO therefore targets a different finite-sample guarantee from PAC-style methods: marginal conformal coverage rather than high-probability risk control. Under consistency assumptions on the coupled risk bound, the two approaches nevertheless converge to the same population threshold. Across classification and regression benchmarks, including ImageNet-A, CIFAR-100, Diabetes, California Housing, and Concrete, DCO tracks the nominal coverage level closely while often reducing average prediction-set size or interval width relative to PAC-style calibration. On ImageNet-A, for example, the average set size decreases from $26.52$ to $25.26$ and the 95th-percentile set size from $58.95$ to $53.73$; on Diabetes, the average interval width decreases from $2.098$ to $1.914$.