Distributed Mean Estimation with Limited Communication (1611.00429v3)

Abstract: Motivated by the need for distributed learning and optimization algorithms with low communication cost, we study communication efficient algorithms for distributed mean estimation. Unlike previous works, we make no probabilistic assumptions on the data. We first show that for $d$ dimensional data with $n$ clients, a naive stochastic binary rounding approach yields a mean squared error (MSE) of $\Theta(d/n)$ and uses a constant number of bits per dimension per client. We then extend this naive algorithm in two ways: we show that applying a structured random rotation before quantization reduces the error to $\mathcal{O}((\log d)/n)$ and a better coding strategy further reduces the error to $\mathcal{O}(1/n)$ and uses a constant number of bits per dimension per client. We also show that the latter coding strategy is optimal up to a constant in the minimax sense i.e., it achieves the best MSE for a given communication cost. We finally demonstrate the practicality of our algorithms by applying them to distributed Lloyd's algorithm for k-means and power iteration for PCA.

• The paper’s main contribution is the development of innovative quantization methods that balance limited communication with reduced mean squared error.
• It employs a random rotation technique to uniformly distribute data variance, achieving an improved error bound of O((log d)/n).
• Variable-length coding is used to further optimize MSE to O(1/n), making the approach highly practical for federated learning and distributed systems.

Distributed Mean Estimation with Limited Communication

In the era of distributed computing, communication efficiency is paramount for the scalability of machine learning algorithms. The paper "Distributed Mean Estimation with Limited Communication" by Suresh et al. addresses a key component of distributed learning systems: the estimation of the mean across multiple clients with restricted communication channels. This work is particularly relevant for applications like federated learning and distributed optimization, where each client may represent a low-bandwidth device such as a mobile phone.

Summary of Contributions

The authors propose several methods to reduce the mean squared error (MSE) in mean estimation while keeping communication costs low. Their approach is notable for not making any probabilistic assumptions about the distributed data, which distinguishes it from previous research. The main contributions include:

1. Naive Stochastic Binary Rounding: Initially, they explore a simple binary quantization scheme where each client sends a constant number of bits per dimension. This approach results in an MSE of $\Theta(d/n)$, where $d$ is the dimensionality of the data, and $n$ is the number of clients.
2. Random Rotation Technique: By introducing a structured random rotation before quantization, the authors improve the error bounds to $\mathcal{O}((\log d)/n)$. This process leverages the randomization of input data to make the variance more uniform across dimensions.
3. Variable-Length Coding: Utilizing a more sophisticated coding strategy, they further improve the MSE to $\mathcal{O}(1/n)$ while maintaining a constant number of bits per dimension per client. The paper argues that this method is optimal up to a constant factor in the minimax sense.
4. Application to Distributed Algorithms: To demonstrate practicality, the proposed algorithms are applied to distributed Lloyd's algorithm for k-means clustering and power iteration methods for Principal Component Analysis (PCA).

Theoretical Implications

The paper's significance lies in its theoretical results, which specify the trade-offs between communication cost and accuracy in distributed mean estimation. By achieving a minimax-optimal MSE with a limited number of bits, the authors provide a roadmap for future developments in efficient distributed systems. Their approach using both random rotations and variable-length coding suggests new avenues for reducing communications overhead without sacrificing estimation accuracy.

Practical Implications

On the practical side, this research underscores the importance of efficient compressive strategies in real-world distributed learning applications. The methods proposed have direct implications for optimizing the runtime and energy consumption of large-scale machine learning frameworks, especially in federated learning environments where clients have limited computational capabilities.

Future Directions

While this paper lays a solid foundation, several directions can be explored further. One such direction could be investigating adaptive quantization strategies that vary with data distribution characteristics to further improve the trade-off between communicational burden and estimation precision. Moreover, exploring the integration of these algorithms with privacy-preserving techniques could make them more applicable in sensitive applications like healthcare and finance.

In summary, the paper offers valuable insights and methodologies that enhance mean estimation in distributed networks under constrained communication settings. Its theoretical insights and practical applications adequately address the communication bottlenecks typical in distributed systems, and they set the stage for future advancements in this rapidly evolving field.