# Tree-Structured Parzen Estimator Can Solve Black-Box Combinatorial Optimization More Efficiently

arXiv: https://arxiv.org/abs/2507.08053 · HTML: https://mlsanity.com/paper/2507.08053/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 37 references that are in the map.
- First posted on arXiv 10 Jul 2025.

## Builds on (most cited first, 22 of 22)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Deep Reinforcement Learning that Matters](https://mlsanity.com/paper/1709.06560/) (2017)
- [BOHB: Robust and Efficient Hyperparameter Optimization at Scale](https://mlsanity.com/paper/1807.01774/) (2018)
- [Tune: A Research Platform for Distributed Model Selection and Training](https://mlsanity.com/paper/1807.05118/) (2018)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer](https://mlsanity.com/paper/2203.03466/) (2022)
- [Tree-Structured Parzen Estimator: Understanding Its Algorithm Components and Their Roles for Better Empirical Performance](https://mlsanity.com/paper/2304.11127/) (2023)
- [Combinatorial Bayesian Optimization using the Graph Cartesian Product](https://mlsanity.com/paper/1902.00448/) (2019)
- [Bayesian Optimization of Combinatorial Structures](https://mlsanity.com/paper/1806.08838/) (2018)
- [On the Importance of Hyperparameter Optimization for Model-based Reinforcement Learning](https://mlsanity.com/paper/2102.13651/) (2021)
- [CMA-ES for Hyperparameter Optimization of Deep Neural Networks](https://mlsanity.com/paper/1604.07269/) (2016)
- [Bounce: Reliable High-Dimensional Bayesian Optimization for Combinatorial and Mixed Spaces](https://mlsanity.com/paper/2307.00618/) (2023)
- [Bayesian Optimization over High-Dimensional Combinatorial Spaces via Dictionary-based Embeddings](https://mlsanity.com/paper/2303.01774/) (2023)
- [On the Importance of Hyperparameters and Data Augmentation for Self-Supervised Learning](https://mlsanity.com/paper/2207.07875/) (2022)
- [Framework and Benchmarks for Combinatorial and Mixed-variable Bayesian Optimization](https://mlsanity.com/paper/2306.09803/) (2023)
- [c-TPE: Tree-structured Parzen Estimator with Inequality Constraints for Expensive Hyperparameter Optimization](https://mlsanity.com/paper/2211.14411/) (2022)
- [Speeding Up Multi-Objective Hyperparameter Optimization by Task Similarity-Based Meta-Learning for the Tree-Structured Parzen Estimator](https://mlsanity.com/paper/2212.06751/) (2022)
- [Multi-Fidelity Methods for Optimization: A Survey](https://mlsanity.com/paper/2402.09638/) (2024)
- [On the Importance of Domain Model Configuration for Automated Planning Engines](https://mlsanity.com/paper/2010.07710/) (2020)
- [Bayesian Variational Optimization for Combinatorial Spaces](https://mlsanity.com/paper/2011.02004/) (2020)
- [Python Wrapper for Simulating Multi-Fidelity Optimization on HPO Benchmarks without Any Wait](https://mlsanity.com/paper/2305.17595/) (2023)
- [Fast Benchmarking of Asynchronous Multi-Fidelity Optimization on Zero-Cost Benchmarks](https://mlsanity.com/paper/2403.01888/) (2024)

## Abstract (arXiv)

Tree-structured Parzen estimator (TPE) is a versatile hyperparameter optimization (HPO) method supported by popular HPO tools. Since these HPO tools have been developed in line with the trend of deep learning (DL), the problem setups often used in the DL domain have been discussed for TPE such as multi-objective optimization and multi-fidelity optimization. However, the practical applications of HPO are not limited to DL, and black-box combinatorial optimization is actively utilized in some domains, e.g., chemistry and biology. As combinatorial optimization has been an untouched, yet very important, topic in TPE, we propose an efficient combinatorial optimization algorithm for TPE. In this paper, we first generalize the categorical kernel with the numerical kernel in TPE, enabling us to introduce a distance structure to the categorical kernel. Then we discuss modifications for the newly developed kernel to handle a large combinatorial search space. These modifications reduce the time complexity of the kernel calculation with respect to the size of a combinatorial search space. In the experiments using synthetic problems, we verified that our proposed method identifies better solutions with fewer evaluations than the original TPE. Our algorithm is available in Optuna, an open-source framework for HPO.