TriEKB-Surg: Triadic Evolutionary Knowledge Bridging for Task-Heterogeneous Federated Surgical Video Understanding

Federated learning (FL) enables institutions to collaboratively train surgical video understanding models without sharing raw data. However, most existing methods assume that all clients optimize the same task, which is unrealistic when institutions provide different forms of supervision. Although heterogeneous-task updates may share the same parameter space and thus be numerically aggregatable, they arise from distinct prediction spaces and optimization objectives. Therefore, parameter aggregatability does not guarantee knowledge transferability: a source update may be semantically irrelevant to the target, unreliable at the current training stage, or incompatible with the target objective. To address this challenge, we propose **Tri**adic **E**volutionary **K**nowledge **B**ridging (**TriEKB-Surg**), which frames task-heterogeneous FL as an evolving and client-specific knowledge-transfer problem. The triadic design considers semantic relevance, optimization-stage reliability, and task-compatible absorption. At the server, dual-prompt compatibility estimates cross-client relevance at coarse task-family and fine task-semantic levels, while an evolutionary curriculum combines source reliability with target-specific update agreement to construct personalized bridges. On the client side, cross-task alignment transforms transferred knowledge into task-compatible auxiliary supervision, while task-aware feature modulation mitigates optimization conflict. By determining *which* knowledge to transfer, *when* to trust it, and *how* to absorb it, TriEKB-Surg enables adaptive collaboration across heterogeneous surgical tasks. Experiments on five public surgical video datasets show that TriEKB-Surg achieves the best overall performance among the evaluated FL methods. Code will be made publicly available.