[Paper Review] Distributed inference for quantile regression processes

Stanislav Volgushev, Shih-Kang Chao+1 more|arXiv (Cornell University)|Jan 21, 2017

Statistical Methods and Inference17 references14 citations

TL;DR

This paper proposes a distributed inference framework for quantile regression processes that enables parallel estimation of conditional quantile functions across multiple computing units, followed by projection-based construction of a full quantile regression process. The method preserves statistical accuracy under optimal choices of computing units and quantile levels, with sharp bounds derived for minimal computational cost.

ABSTRACT

The increased availability of massive data sets provides a unique opportunity to discover subtle patterns in their distributions, but also imposes overwhelming computational challenges. To fully utilize the information contained in big data, we propose a two-step procedure: (i) estimate conditional quantile functions at different levels in a parallel computing environment; (ii) construct a conditional quantile regression process through projection based on these estimated quantile curves. Our general quantile regression framework covers both linear models with fixed or growing dimension and series approximation models. We prove that the proposed procedure does not sacrifice any statistical inferential accuracy provided that the number of distributed computing units and quantile levels are chosen properly. In particular, a sharp upper bound for the former and a sharp lower bound for the latter are derived to capture the minimal computational cost from a statistical perspective. As an important application, the statistical inference on conditional distribution functions is considered. Moreover, we propose computationally efficient approaches to conducting inference in the distributed estimation setting described above. Those approaches directly utilize the availability of estimators from sub-samples and can be carried out at almost no additional computational cost. Simulations confirm our statistical inferential theory.

Motivation & Objective

• Address the computational challenges of analyzing massive datasets in quantile regression.
• Develop a scalable framework that maintains statistical accuracy despite distributed computation.
• Establish theoretical bounds on the number of computing units and quantile levels required for optimal performance.
• Enable efficient statistical inference on conditional distribution functions using distributed estimators.
• Minimize additional computational cost in inference by leveraging existing sub-sample estimators.

Proposed method

• Estimate conditional quantile functions at multiple quantile levels in parallel across distributed computing units.
• Construct a conditional quantile regression process via projection using the distributedly estimated quantile curves.
• Formulate the framework to support both fixed/growing-dimensional linear models and series approximation models.
• Derive sharp upper bounds on the number of computing units and sharp lower bounds on the number of quantile levels to ensure statistical efficiency.
• Utilize estimators from sub-samples directly for inference, enabling near-zero additional computational cost.
• Apply projection-based methods to combine distributed estimates into a coherent quantile regression process.

Experimental results

Research questions

• RQ1 What is the minimal number of computing units required to maintain statistical accuracy in distributed quantile regression?
• RQ2 What is the minimal number of quantile levels needed to ensure no loss of inferential accuracy?
• RQ3 How can inference on conditional distribution functions be efficiently conducted in a distributed setting?
• RQ4 Can statistical inference be performed with negligible additional computational cost after distributed estimation?
• RQ5 Does the proposed method preserve the same inferential accuracy as centralized estimation?

Key findings

• The proposed distributed inference procedure maintains the same statistical inferential accuracy as centralized estimation when the number of computing units and quantile levels are chosen appropriately.
• A sharp upper bound is derived for the number of computing units required to avoid loss of efficiency.
• A sharp lower bound is established for the number of quantile levels to ensure no statistical cost is incurred.
• The method enables efficient inference on conditional distribution functions using only the distributed estimators, with almost no additional computational cost.
• Simulations confirm the theoretical findings, demonstrating that the method achieves accurate inference under the derived bounds.
• The framework is applicable to both linear models with fixed or growing dimensions and series approximation models.