# Symbol: Generating Flexible Black-Box Optimizers through Symbolic Equation Learning

arXiv: https://arxiv.org/abs/2402.02355 · HTML: https://mlsanity.com/paper/2402.02355/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 42 references that are in the map.
- First posted on arXiv 4 Feb 2024.

## Builds on (most cited first, 16 of 16)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Scalable Bayesian Optimization Using Deep Neural Networks](https://mlsanity.com/paper/1502.05700/) (2015)
- [SMAC3: A Versatile Bayesian Optimization Package for Hyperparameter Optimization](https://mlsanity.com/paper/2109.09831/) (2021)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [HPO-B: A Large-Scale Reproducible Benchmark for Black-Box HPO based on OpenML](https://mlsanity.com/paper/2106.06257/) (2021)
- [SymFormer: End-to-end symbolic regression using transformer-based architecture](https://mlsanity.com/paper/2205.15764/) (2022)
- [Discovering Evolution Strategies via Meta-Black-Box Optimization](https://mlsanity.com/paper/2211.11260/) (2022)
- [Deep Reinforcement Learning Based Parameter Control in Differential Evolution](https://mlsanity.com/paper/1905.08006/) (2019)
- [Discovering Attention-Based Genetic Algorithms via Meta-Black-Box Optimization](https://mlsanity.com/paper/2304.03995/) (2023)
- [Learning adaptive differential evolution algorithm from optimization experiences by policy gradient](https://mlsanity.com/paper/2102.03572/) (2021)
- [Learning to Optimize in Swarms](https://mlsanity.com/paper/1911.03787/) (2019)
- [Meta-Learning for Black-box Optimization](https://mlsanity.com/paper/1907.06901/) (2019)

## Built on by (most cited first, 22 of 22)

- [Toward Automated Algorithm Design: A Survey and Practical Guide to Meta-Black-Box-Optimization](https://mlsanity.com/paper/2411.00625/) (2024)
- [Auto-configuring Exploration-Exploitation Tradeoff in Evolutionary Computation via Deep Reinforcement Learning](https://mlsanity.com/paper/2404.08239/) (2024)
- [ConfigX: Modular Configuration for Evolutionary Algorithms via Multitask Reinforcement Learning](https://mlsanity.com/paper/2412.07507/) (2024)
- [Pretrained Optimization Model for Zero-Shot Black Box Optimization](https://mlsanity.com/paper/2405.03728/) (2024)
- [Reinforcement Learning-based Self-adaptive Differential Evolution through Automated Landscape Feature Learning](https://mlsanity.com/paper/2503.18061/) (2025)
- [Generative Discovery of Partial Differential Equations by Learning from Math Handbooks](https://mlsanity.com/paper/2505.05869/) (2025)
- [A Systematic Survey on Large Language Models for Evolutionary Optimization: From Modeling to Solving](https://mlsanity.com/paper/2509.08269/) (2025)
- [Neural Exploratory Landscape Analysis for Meta-Black-Box-Optimization](https://mlsanity.com/paper/2408.10672/) (2024)
- [MetaBox-v2: A Unified Benchmark Platform for Meta-Black-Box Optimization](https://mlsanity.com/paper/2505.17745/) (2025)
- [RLEMMO: Evolutionary Multimodal Optimization Assisted By Deep Reinforcement Learning](https://mlsanity.com/paper/2404.08242/) (2024)
- [Surrogate Learning in Meta-Black-Box Optimization: A Preliminary Study](https://mlsanity.com/paper/2503.18060/) (2025)
- [LLM-Meta-SR: In-Context Learning for Evolving Selection Operators in Symbolic Regression](https://mlsanity.com/paper/2505.18602/) (2025)
- [Instance Generation for Meta-Black-Box Optimization through Latent Space Reverse Engineering](https://mlsanity.com/paper/2509.15810/) (2025)
- [DesignX: Human-Competitive Algorithm Designer for Black-Box Optimization](https://mlsanity.com/paper/2505.17866/) (2025)
- [Advancing CMA-ES with Learning-Based Cooperative Coevolution for Scalable Optimization](https://mlsanity.com/paper/2504.17578/) (2025)
- [LiBOG: Lifelong Learning for Black-Box Optimizer Generation](https://mlsanity.com/paper/2505.13025/) (2025)
- [Towards certifiable AI in aviation: landscape, challenges, and opportunities](https://mlsanity.com/paper/2409.08666/) (2024)
- [TRACE: A Generalizable Drift Detector for Streaming Data-Driven Optimization](https://mlsanity.com/paper/2512.07082/) (2025)
- [Task-free Adaptive Meta Black-box Optimization](https://mlsanity.com/paper/2601.21475/) (2026)
- [Meta-Learning-Assisted Constraint Relaxation for Constrained Black-Box Optimization](https://mlsanity.com/paper/2602.00532/) (2026)
- [Meta-Black-Box Optimization with Ensemble Surrogate Modeling for Robustness-Accuracy Trade-off within SAEA](https://mlsanity.com/paper/2606.00862/) (2026)
- [READY: Reward Discovery for Meta-Black-Box Optimization](https://mlsanity.com/paper/2601.21847/) (2026)

## Abstract (arXiv)

Recent Meta-learning for Black-Box Optimization (MetaBBO) methods harness neural networks to meta-learn configurations of traditional black-box optimizers. Despite their success, they are inevitably restricted by the limitations of predefined hand-crafted optimizers. In this paper, we present textscSymbol, a novel framework that promotes the automated discovery of black-box optimizers through symbolic equation learning. Specifically, we propose a Symbolic Equation Generator (SEG) that allows closed-form optimization rules to be dynamically generated for specific tasks and optimization steps. Within textscSymbol, we then develop three distinct strategies based on reinforcement learning, so as to meta-learn the SEG efficiently. Extensive experiments reveal that the optimizers generated by textscSymbol not only surpass the state-of-the-art BBO and MetaBBO baselines, but also exhibit exceptional zero-shot generalization abilities across entirely unseen tasks with different problem dimensions, population sizes, and optimization horizons. Furthermore, we conduct in-depth analyses of our textscSymbol framework and the optimization rules that it generates, underscoring its desirable flexibility and interpretability.