Bulyan is a meta-aggregation rule that combines Krum selection with coordinate-wise trimmed mean — first using Multi-Krum to select the most trustworthy subset of $ heta$ clients, then applying trimmed mean on this selected subset for an extra layer of robustness.

How Bulyan Works

• Step 1 (Krum Selection): Use Multi-Krum to select the top-$ heta$ most central client updates ($ heta = n - 2f$).
• Step 2 (Trimmed Mean): Apply coordinate-wise trimmed mean on the $ heta$ selected updates (trim $f$ from each side).
• Double Filter: Byzantine updates must survive both Krum distance-based filtering AND trimmed mean outlier removal.
• Robustness: Tolerates $f < (n-3)/4$ Byzantine clients with strong guarantees.

Why It Matters

• Stronger Than Either: Bulyan is more robust than either Krum or trimmed mean alone — double filtering.
• Dimensional Attacks: Defends against attacks that exploit the weakness of coordinate-wise methods.
• Trade-Off: Requires more honest clients ($n > 4f + 3$) — stronger requirement than simple median or Krum.

Bulyan is the double-filtered aggregation — using Krum to vet clients, then trimmed mean to clean their updates, for maximum Byzantine robustness.