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 \textsc{Symbol}, 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 \textsc{Symbol}, 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 \textsc{Symbol} 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 \textsc{Symbol} framework and the optimization rules that it generates, underscoring its desirable flexibility and interpretability.

翻译：近期，针对黑箱优化的元学习方法（MetaBBO）利用神经网络元学习传统黑箱优化器的配置。尽管取得了成功，但它们不可避免地受到预定义手工优化器局限性的制约。本文提出\textsc{Symbol}——一种通过符号方程学习自动发现黑箱优化器的新框架。具体而言，我们设计了符号方程生成器（SEG），能够针对特定任务和优化步骤动态生成闭式优化规则。在\textsc{Symbol}框架内，我们进一步开发了三种基于强化学习的差异化策略，以实现对SEG的高效元学习。大量实验表明，\textsc{Symbol}生成的优化器不仅超越了最先进的BBO和MetaBBO基线，还在具有不同问题维度、种群规模以及优化时域的全新任务上展现出卓越的零样本泛化能力。此外，我们对\textsc{Symbol}框架及其生成的优化规则进行了深入分析，突显了其理想的灵活性与可解释性。