Under distributed differential privacy, the minimax error of federated PCA at the central server is the harmonic mean of the local clients' minimax errors, up to log factors.
Using these facts, we have E X j1,··· ,jl∈[m] wj1 · · ·wjltr(U ⊤M (s1)M (s1)⊤∆j1M (s2) · · ·M (sl)⊤∆jlM (sl+1)M (sl+1)⊤U ) · 1(E) ≤ r · X j∈I2(s) uj1 · · ·ujl
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Federated PCA and Estimation for Spiked Covariance Matrices: Optimal Rates and Efficient Algorithm
Under distributed differential privacy, the minimax error of federated PCA at the central server is the harmonic mean of the local clients' minimax errors, up to log factors.