Does Execution Order Imply Causality? Federated Post-Training of Asynchronous LLM Agents over Execution Posets

Language model agents often execute multiple tools and subtasks concurrently, while the resulting traces remain distributed across private clients. Federated adaptation allows these clients to improve a shared agent without centralizing their execution data. However, existing methods usually convert each execution into a single token sequence. In asynchronous environments, independent events may finish in different orders because of tool latency, queueing, or local scheduling. As a result, two clients can complete the same execution, observe the same tool outputs, and make the same decision, yet store the intermediate events in different orders. Sequential training treats these order differences as different positions and contexts, which produces different local gradients from the same causal execution. We call this problem interleaving heterogeneity. To address it, we propose TopoFed-Agent, which represents each execution as a partial order certified by runtime evidence. During local adaptation, each event attends only to its causal ancestors, uses a position determined by its causal rank, and receives supervision attached to the event itself. In this way, changing the legal storage order of independent events does not change the learning problem. We prove that, for a fixed certified execution, all legal serializations produce identical predictions, losses, and parameter gradients. Therefore, gradient differences caused only by client schedulers are eliminated. Experiments across four execution structures and five Transformer initializations show that sequential training yields relative full-parameter gradient deviations ranging from 0.26 to 0.46 across legal serializations, whereas TopoFed-Agent keeps the deviation below \(2.31\times10^-16\).