Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality (1506.07216v3)

Abstract: We study the tradeoff between the statistical error and communication cost of distributed statistical estimation problems in high dimensions. In the distributed sparse Gaussian mean estimation problem, each of the $m$ machines receives $n$ data points from a $d$-dimensional Gaussian distribution with unknown mean $\theta$ which is promised to be $k$-sparse. The machines communicate by message passing and aim to estimate the mean $\theta$. We provide a tight (up to logarithmic factors) tradeoff between the estimation error and the number of bits communicated between the machines. This directly leads to a lower bound for the distributed \textit{sparse linear regression} problem: to achieve the statistical minimax error, the total communication is at least $\Omega(\min{n,d}m)$, where $n$ is the number of observations that each machine receives and $d$ is the ambient dimension. These lower results improve upon [Sha14,SD'14] by allowing multi-round iterative communication model. We also give the first optimal simultaneous protocol in the dense case for mean estimation. As our main technique, we prove a \textit{distributed data processing inequality}, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems.

• The paper provides theoretical lower bounds on communication required to achieve statistical minimax errors in distributed high-dimensional estimation, showing specific bounds for sparse Gaussian mean estimation.
• It introduces a novel distributed data processing inequality, a mathematical tool for rigorously deriving these communication lower bounds in distributed settings.
• The paper also shows that communication costs scale with dimensionality for sparse linear regression and proposes an optimal protocol for dense Gaussian mean estimation.

Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality

The paper "Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality" authored by Mark Braverman, Ankit Garg, Tengyu Ma, Huy L. Nguyen, and David P. Woodruff, explores the intricate tradeoffs between statistical error and communication cost in distributed setups for statistical estimation tasks in high-dimensional spaces. It focuses primarily on two problems: distributed sparse Gaussian mean estimation and distributed sparse linear regression.

The core research problem addressed in this work concerns the inherent limits on communication efficiencies when estimating statistical parameters, particularly in high-dimensional contexts. The distributed environment, where data exchange between machines is conducted through message passing, has communication as a critical bottleneck. The paper identifies the parameters influencing this bottleneck and sets theoretical lower bounds for communication complexity required to achieve certain levels of statistical accuracy.

Key Contributions

1. Communication-Error Tradeoff: The authors provide a theoretical framework that sets a lower bound on the communication necessary for reaching statistical minimax errors in distributed scenarios. Specifically, for sparse Gaussian mean estimation, the lower bound on communication is derived as Ω(min{n,d}m) to achieve the statistical minimax error, where n is the number of observations per machine, d is the dimension, and m is the number of machines.
2. Distributed Data Processing Inequality: The paper introduces a novel distributed data processing inequality that extends traditional data processing inequalities to distributed scenarios. This mathematical formulation allows the authors to rigorously deduce lower bounds on the communication cost necessary for achieving specific error rates in estimation tasks.
3. Sparse Linear Regression: For sparse linear regression, the authors prove that the communication cost must scale with dimensionality, even when the sparsity structure of the parameter is directly exploitative, indicating that high-dimensional vectors are essential for accurate estimation.
4. Optimal Protocol for Dense Gaussian Estimation: Along with lower bounds, the paper also proposes an optimal protocol for dense Gaussian mean estimation in the simultaneous communication model. This provides a practical solution and benchmarks the efficiency protocols against theoretical lower bounds.

Implications and Future Directions

The implications of this work are broad and impactful. By laying the ground for understanding the fundamental limits of communication in distributed learning systems, it paves the way for developing more efficient algorithms that can approach these theoretical lower bounds.

From a theoretical standpoint, the distributed data processing inequality could find applications in other areas where distributed learning and estimation are pivotal, such as federated learning, distributed sensor networks, and large-scale data analytics.

Looking forward, potential further developments in this field may involve:

• Extending these bounds to non-linear models and other distribution families.
• Exploring the role of privacy and encoding strategies in optimizing communication.
• Application of these results in real-world distributed system architectures to fine-tune communication protocols for efficiency.

Conclusion

This paper significantly advances the understanding of communication complexities in distributed statistical estimation, providing new mathematical tools like the distributed data processing inequality and offering practical benchmarking protocols for dense Gaussian mean estimation. As distributed systems and machine learning continue to intersect, these insights provide a crucial foundation upon which future advancements can be built.