Despite the potential of federated learning in medical applications, inconsistent imaging quality across institutions-stemming from lower-quality data from a minority of clients-biases federated models toward more common high-quality images. This raises significant fairness concerns. Existing fair federated learning methods have demonstrated some effectiveness in solving this problem by aligning a single 0th- or 1st-order state of convergence (e.g., training loss or sharpness). However, we argue in this work that fairness based on such a single state is still not an adequate surrogate for fairness during testing, as these single metrics fail to fully capture the convergence characteristics, making them suboptimal for guiding fair learning. To address this limitation, we develop a generalized framework. Specifically, we propose assessing convergence using multiple states, defined as sharpness or perturbed loss computed at varying search distances. Building on this comprehensive assessment, we propose promoting fairness for these states across clients to achieve our ultimate fairness objective. This is accomplished through the proposed method, FedISM+. In FedISM+, the search distance evolves over time, progressively focusing on different states. We then incorporate two components in local training and global aggregation to ensure cross-client fairness for each state. This gradually makes convergence equitable for all states, thereby improving fairness during testing. Our empirical evaluations, performed on the well-known RSNA ICH and ISIC 2019 datasets, demonstrate the superiority of FedISM+ over existing state-of-the-art methods for fair federated learning. The code is available at https://github.com/wnn2000/FFL4MIA.

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