krum aggregation

**Krum** is a **Byzantine-robust aggregation rule for federated learning that selects the single client update closest to its nearest neighbors** — rather than averaging all updates, Krum picks the one that is most consistent with the majority of other updates. **How Krum Works** - **Distances**: For each client $i$, compute the sum of distances to its $n - f - 2$ closest neighbors. - **Selection**: Choose the client $i^*$ with the smallest sum of neighbor distances — the "most central" update. - **Update**: Use $g_{i^*}$ as the aggregated gradient (single-point selection, not an average). - **Robustness**: Tolerates $f < (n-2)/2$ Byzantine clients. **Why It Matters** - **Geometric**: Krum uses the geometric structure of gradient space — selects the densest cluster. - **One-Shot**: No iterative computation — just distance calculations and a minimum selection. - **Limitation**: Single selection has high variance — Multi-Krum addresses this. **Krum** is **pick the most agreeable update** — selecting the client whose gradient is most consistent with the majority for Byzantine-robust aggregation.