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

TL;DR

This paper tackles the challenge of designing flexible black-box optimizers by learning symbolic, closed-form update rules. It introduces the Symbol framework, featuring a Symbolic Equation Generator (SEG) that autoregressively constructs update rules from a basis symbol set, and three RL-based training strategies (Symbol-E, Symbol-G, Symbol-S) to optimize SEG across diverse BBO tasks in a bi-level MetaBBO setting. Empirical results show state-of-the-art performance and robust zero-shot generalization on unseen tasks, including hyper-parameter optimization and protein docking, with interpretability through analyzed update rules. The approach offers a flexible, interpretable alternative to hand-crafted optimizers and end-to-end black-box policies, with practical impact for rapid, generalizable optimization across domains.

Abstract

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.

Figures (10)

• Figure 1: Comparison of traditional BBO, MetaBBO, and our Symbol frameworks. Left: Depiction of various BBO methods. Right: A view of the bi-level structure of Symbol. The lower level optimizes individual BBO tasks, with the SEG generating step-by-step update rules. The meta level compiles performances from the lower level into a meta objective so as to learn the SEG through either the Symbol-E (without a teacher BBO optimizer), Symbol-G, or Symbol-S strategy.
• Figure 2: The illustration of our symbolic equation generator (SEG) at the lower level. Left: An example of generating the update rule $\tau$ via SEG with on-the-fly inference of constant $c$ (highlighted in red dashed block). Top right: The symbolic expression tree from the pre-order traversal and the interpreted update rule $\tau$. Bottom right: an illustration of the overall iterations of SEG.
• Figure 3: Performance of three different training strategies.
• Figure 4: Evolution visulization of the optimizers, showing the position of population (red dots) and the global optimal (yellow star). Top: Symbol-S; Middle: original DE; Bottom: CMA-ES.
• Figure 5: Performance of Symbol-S with different teachers.