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Fundamental Limits of Distributed Covariance Matrix Estimation Under Communication Constraints
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Derives a min-max lower bound and an achievable scheme for distributed covariance estimation where two agents hold disjoint dimensions of sub-Gaussian samples and send bit-limited messages to a server, matching up to a log factor in homogeneous regimes.
Problem
Estimating a high-dimensional covariance matrix at a central server when two agents each observe a disjoint block of dimensions (vertical split) and can send only a limited number of bits, with finitely many samples.
Approach
Two-part analysis: a min-max lower bound via Fano's method combined with strong data processing inequalities for Gaussian families (one varying cross-covariance, one varying self-covariance), and an achievable scheme that quantizes empirical self-covariances and blocks of raw data with vector quantizers, reconstructing the cross-covariance at the server.
Key results
- Lower bound Omega(max{sqrt(d1d2/Bmin), sqrt(d/m), 2^{-min(B1/d1^2, B2/d2^2)}}) on operator-norm error for any estimator.
- Achievable scheme with error O~(sqrt(d/m) + sqrt(max{d1^2/B1, d2^2/B2, d1d2/Bmin})), matching the lower bound up to a log factor in eta-homogeneous regimes.
- Shows that the naive extension of Hadar et al. 2019 gives a weaker bound by a factor of d_min on the Frobenius cross-covariance term.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed communication-constrained covariance estimation scheme | DCME · DCME(σ, m, d1:2, B1:2), under the Theorem 4.4 assumptions | expected operator-norm distortion: σ²O(√(d/m) + log₂(Bmin/(d1d2))√max{d1²/B1, d2²/B2, d1d2/Bmin}) | Helped |
| Use the proposed scheme in an η-homogeneous setting | DCME · η-homogeneous DCME | expected operator-norm distortion: σ²O(√(d/m) + log₂(Bmin/(d1d2))√(η d1d2/Bmin)) | Helped |
| Use the proposed scheme instead of the lower-bound-limited optimum | DCME in the η-homogeneous setting · η-homogeneous DCME | expected operator-norm distortion: up to a logarithmic term | Mixed |
| Use the naїve multivariate BCR extension for cross-covariance estimation instead of the paper's lower-bound method | Distributed covariance estimation with d1 > 1 and d2 > 1 · Cross-covariance matrix estimation | Frobenius-norm distortion lower bound: by a factor of dmin | Hurt |
Surprising findings
- To achieve a satisfactory approximation, the communication budget of the poor agent must scale in proportion to the dimension of the rich agent.
Limitations
- Achievability matches the lower bound only up to a log factor and only in homogeneous (eta-homogeneous) regimes.
- Analysis restricted to two agents and sub-Gaussian data with a finite number of samples.
Experimental setup
Metrics: expected operator norm error expected Frobenius norm error
Baselines: Hadar et al. 2019 method extension Hadar & Shayevitz 2019 method extension
Builds on: Hadar et al. 2019 communication complexity of estimating correlations Hadar & Shayevitz 2019 distributed estimation of Gaussian correlations Duchi Fano method lecture notes
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