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Collaborative Compressors in Distributed Mean Estimation with Limited Communication Budget

Harsh Vardhan, Arya Mazumdar

TL;DR

The paper tackles distributed mean estimation of high-dimensional vectors under limited communication by introducing four collaborative compressors that do not require prior correlation knowledge. It develops HadamardMultiDim and SparseReg to achieve strong $\ell_\infty$ and $\ell_2$ error decay with the number of clients, while NoisySign and OneBit address unbounded coordinates and cosine-distance objectives, respectively. Theoretical analyses tie estimation error to inter-client dissimilarities and demonstrate graceful degradation as vectors diverge, with empirical results showing improved performance in low-dissimilarity regimes and applicability to downstream distributed tasks. Collectively, the work advances communication-efficient DME through agnostic collaboration across multiple error metrics, offering practical schemes with provable guarantees.

Abstract

Distributed high dimensional mean estimation is a common aggregation routine used often in distributed optimization methods. Most of these applications call for a communication-constrained setting where vectors, whose mean is to be estimated, have to be compressed before sharing. One could independently encode and decode these to achieve compression, but that overlooks the fact that these vectors are often close to each other. To exploit these similarities, recently Suresh et al., 2022, Jhunjhunwala et al., 2021, Jiang et al, 2023, proposed multiple correlation-aware compression schemes. However, in most cases, the correlations have to be known for these schemes to work. Moreover, a theoretical analysis of graceful degradation of these correlation-aware compression schemes with increasing dissimilarity is limited to only the $\ell_2$-error in the literature. In this paper, we propose four different collaborative compression schemes that agnostically exploit the similarities among vectors in a distributed setting. Our schemes are all simple to implement and computationally efficient, while resulting in big savings in communication. The analysis of our proposed schemes show how the $\ell_2$, $\ell_\infty$ and cosine estimation error varies with the degree of similarity among vectors.

Collaborative Compressors in Distributed Mean Estimation with Limited Communication Budget

TL;DR

The paper tackles distributed mean estimation of high-dimensional vectors under limited communication by introducing four collaborative compressors that do not require prior correlation knowledge. It develops HadamardMultiDim and SparseReg to achieve strong and error decay with the number of clients, while NoisySign and OneBit address unbounded coordinates and cosine-distance objectives, respectively. Theoretical analyses tie estimation error to inter-client dissimilarities and demonstrate graceful degradation as vectors diverge, with empirical results showing improved performance in low-dissimilarity regimes and applicability to downstream distributed tasks. Collectively, the work advances communication-efficient DME through agnostic collaboration across multiple error metrics, offering practical schemes with provable guarantees.

Abstract

Distributed high dimensional mean estimation is a common aggregation routine used often in distributed optimization methods. Most of these applications call for a communication-constrained setting where vectors, whose mean is to be estimated, have to be compressed before sharing. One could independently encode and decode these to achieve compression, but that overlooks the fact that these vectors are often close to each other. To exploit these similarities, recently Suresh et al., 2022, Jhunjhunwala et al., 2021, Jiang et al, 2023, proposed multiple correlation-aware compression schemes. However, in most cases, the correlations have to be known for these schemes to work. Moreover, a theoretical analysis of graceful degradation of these correlation-aware compression schemes with increasing dissimilarity is limited to only the -error in the literature. In this paper, we propose four different collaborative compression schemes that agnostically exploit the similarities among vectors in a distributed setting. Our schemes are all simple to implement and computationally efficient, while resulting in big savings in communication. The analysis of our proposed schemes show how the , and cosine estimation error varies with the degree of similarity among vectors.
Paper Structure (30 sections, 8 theorems, 40 equations, 3 figures, 2 tables, 6 algorithms)

This paper contains 30 sections, 8 theorems, 40 equations, 3 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Notation.
  3. Organization.
  4. Related Works
  5. NoisySign for unbounded $\lvert\lvert g_i\rvert\rvert_\infty$
  6. Optimal Dependence on $m$
  7. HadamardMultiDim
  8. Sparse regression coding
  9. Benefits of HadmardMultiDim and SparseReg
  10. One-bit Schemes
  11. Comparison to $\ell_2$ compressor.
  12. Experiments
  13. Conclusion
  14. Acknowledgment.
  15. Proofs for Section \ref{['sec:noisy_sign']}
  16. ...and 15 more sections

Key Result

Theorem 1

With probability $1-2dm^{-c}$, for some constant $c>0$, the estimation error of alg:noisy_sign is where $\Delta_\Phi \triangleq \max_{j\in [d]}\lvert \frac{1}{m} \sum_{i=1}^m \Phi_\sigma(g_i^{(j)}) - \Phi_\sigma(g^{(j)}) \rvert$ and $\alpha(u) \triangleq 1 - \Phi_\sigma(u)$. Further, for small $\Delta_{\phi}$ and $\frac{\lvert\lvert g\rvert\rvert_\infty}{\sigma}$, and large $m$,

Figures (3)

  • Figure 1: Compression for Distributed Mean Estimation
  • Figure 2: Performance of DME(Distributed Mean Estimation), KMeans, Power iteration and linear regression for the same communication budget. For each experiment, we report the best compressors. Lin. Reg. refer to Linear Regression. For power iteration, higher top eigenvalue is better. For all other experiments, we report the error, so lower is better.
  • Figure 3: Performance of compressors for Logistic regression on HAR har dataset. Left: Training Logistic loss, Right : Test Accuracy.

Theorems & Definitions (11)

  • Theorem 1: Estimation error of noisy sign
  • Theorem 2: HadamardMultiDim Error
  • Definition 1: $(\delta_1, \delta_2, \Gamma)$-Cover
  • Remark 1: Gaussian $A$
  • Theorem 3: SparseReg Error
  • Remark 2: Growth of $\Lambda$ with $m$
  • Lemma 1: Malicious Noise
  • Theorem 4: Error of Technique I
  • Theorem 5: Error of Technique II
  • Corollary 1
  • ...and 1 more