This research introduces Refined-ZO algorithms that significantly improve optimization when gradient information is unavailable or too costly to compute.

• Enhances optimization by better utilizing moment information during the process
• Achieves faster convergence compared to existing zeroth-order methods
• Particularly valuable for black-box systems and resource-constrained environments

For engineering applications, these advancements enable more efficient optimization of complex systems where internal mechanisms are inaccessible, such as proprietary software, hardware interfaces, or third-party APIs.

Refining Adaptive Zeroth-Order Optimization at Ease