Federated Optimal Transport

The core mechanism of FOT computes the minimal cost of transforming one client's data distribution into another's. Unlike KL-divergence or JS-divergence, the Wasserstein metric respects the underlying geometry of the feature space, providing meaningful gradients even when distributions have non-overlapping support—a common scenario in medical imaging where different hospitals use different scanner vendors. The Kantorovich relaxation enables efficient computation of the optimal coupling matrix between empirical distributions.

FOT aligns latent feature representations across clients without exchanging raw data. Each client computes a local optimal transport map to a shared barycenter distribution—a Wasserstein average of all client distributions. Only the aligned representations or transport plans are communicated, preserving patient privacy. This approach naturally integrates with differential privacy guarantees by adding calibrated noise to the shared barycenter updates, satisfying HIPAA and GDPR requirements for multi-institutional studies.

In clinical federated networks, label distribution skew is pervasive—one hospital may specialize in oncology while another handles primarily cardiology cases. FOT addresses this by computing class-conditional optimal transport maps that align feature distributions separately for each diagnostic category. This prevents the model from erroneously matching a tumor feature from Hospital A to a healthy tissue feature from Hospital B, maintaining semantic consistency during the alignment process.

Traditional optimal transport scales quadratically with sample size, making it prohibitive for large clinical datasets. Mini-batch OT approximates the full transport plan by computing couplings on randomly sampled subsets, reducing complexity from O(n²) to O(b²) where b is batch size. Combined with entropic regularization via the Sinkhorn algorithm, this enables efficient GPU-accelerated alignment during each federated training round without sacrificing convergence guarantees.

FOT learns transport-invariant feature representations that generalize to entirely unseen hospitals. By training a feature extractor to minimize the Wasserstein distance between all source client distributions simultaneously, the model discovers domain-agnostic biomarkers. A model trained on data from three hospitals using FOT can deploy to a fourth hospital with a different CT scanner model and achieve comparable diagnostic accuracy without any local fine-tuning, addressing the critical challenge of scanner-induced batch effects.

The Wasserstein barycenter serves as the central reference distribution in FOT, computed iteratively across clients without centralizing data. Each client computes its local transport map to the current barycenter estimate, then the server aggregates these maps to update the barycenter. This process converges to the Fréchet mean of all client distributions under the Wasserstein metric. The barycenter provides a natural initialization point for new clients joining the federation, enabling plug-and-play scalability in growing healthcare networks.

The fundamental metric minimized by optimal transport, also known as the Earth Mover's Distance. It quantifies the minimal cost to transform one probability distribution into another.

• Measures the geometric discrepancy between distributions
• Unlike KL divergence, provides meaningful gradients even when distributions have non-overlapping supports
• In federated settings, serves as the alignment objective between client feature representations

An entropy-regularized approximation of optimal transport that makes the computation tractable for high-dimensional federated learning scenarios.

• Adds an entropic penalty to smooth the transport plan
• Reduces computational complexity from O(n³ log n) to near-linear time
• Enables differentiable transport plans suitable for gradient-based optimization in neural networks

The application of optimal transport to align feature representations across different client domains, directly addressing feature distribution skew in federated networks.

• Maps source domain samples to target domain through a learned transport plan
• Preserves the geometric structure of each domain during alignment
• Often combined with adversarial training for robust invariant representations

A class of techniques that align local client distributions to a common reference distribution before or during aggregation, preventing client drift.

• Uses barycenter computation to find the Wasserstein mean of all client distributions
• Can operate in feature space or output logit space
• Particularly effective when label distribution skew causes divergent local optima

An extension of optimal transport that compares distributions defined in different metric spaces, making it ideal for heterogeneous federated networks.

• Aligns distributions by comparing intra-domain distances rather than direct sample matching
• Enables transport between clients with different feature dimensionalities
• Critical for federated multi-modal fusion where data modalities differ across institutions

Stochastic optimization strategies that compute optimal transport couplings on mini-batches rather than full datasets, enabling scalability to large clinical data silos.

• Uses sub-sampling to estimate transport plans with bounded error
• Compatible with online learning scenarios where data arrives continuously
• Reduces memory footprint from O(n²) to O(b²) where b is batch size