median aggregation

**Median Aggregation** is a **Byzantine-robust aggregation rule for federated learning that takes the coordinate-wise median of client updates** — for each gradient coordinate, the median value across all clients is selected, making the aggregation resilient to outlier or adversarial updates. **Median Aggregation Details** - **Coordinate-Wise**: For each dimension $i$: $hat{g}_i = ext{median}(g_{1,i}, g_{2,i}, ldots, g_{n,i})$. - **Robustness**: Tolerates up to $f < n/2$ Byzantine clients — the median is determined by the honest majority. - **Geometric Median**: Alternative — find the point minimizing the sum of distances to all updates (considers dimension correlations). - **Computational**: Coordinate-wise median is $O(n log n)$ per dimension. Geometric median requires iterative optimization. **Why It Matters** - **Simple and Effective**: Drop-in replacement for simple averaging — just change mean to median. - **Breakdown Point**: The median has a breakdown point of 50% — can tolerate up to half the values being adversarial. - **Baseline**: Often used as the baseline robust aggregation method for comparison. **Median Aggregation** is **the majority vote for gradients** — selecting the middle value to ignore extreme outliers from malicious or faulty clients.