Language-based agentic systems have shown great promise in recent years, transitioning from solving small-scale research problems to being deployed in challenging real-world tasks. However, optimizing these systems often requires substantial manual labor. Recent studies have demonstrated that these systems can be represented as computational graphs, enabling automatic optimization. Despite these advancements, most current efforts in Graph-based Agentic System Optimization (GASO) fail to properly assign feedback to the system's components given feedback on the system's output. To address this challenge, we formalize the concept of semantic backpropagation with semantic gradients -- a generalization that aligns several key optimization techniques, including reverse-mode automatic differentiation and the more recent TextGrad by exploiting the relationship among nodes with a common successor. This serves as a method for computing directional information about how changes to each component of an agentic system might improve the system's output. To use these gradients, we propose a method called semantic gradient descent which enables us to solve GASO effectively. Our results on both BIG-Bench Hard and GSM8K show that our approach outperforms existing state-of-the-art methods for solving GASO problems. A detailed ablation study on the LIAR dataset demonstrates the parsimonious nature of our method. A full copy of our implementation is publicly available at https://github.com/HishamAlyahya/semantic_backprop
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