# Quantile Regression Under Memory Constraint

arXiv: https://arxiv.org/abs/1810.08264 · HTML: https://mlsanity.com/paper/1810.08264/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2018.

## Built on by (most cited first, 17 of 17)

- [Variance Reduced Median-of-Means Estimator for Byzantine-Robust Distributed Inference](https://mlsanity.com/paper/2103.02860/) (2021)
- [Communication-Constrained Distributed Quantile Regression with Optimal Statistical Guarantees](https://mlsanity.com/paper/2110.13113/) (2021)
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- [Fast and Robust Sparsity Learning over Networks: A Decentralized Surrogate Median Regression Approach](https://mlsanity.com/paper/2202.05498/) (2022)
- [A Piecewise Lyapunov Analysis of Sub-quadratic SGD: Applications to Robust and Quantile Regression](https://mlsanity.com/paper/2504.08178/) (2025)
- [Distributed High-Dimensional Quantile Regression: Estimation Efficiency and Support Recovery](https://mlsanity.com/paper/2405.07552/) (2024)
- [Decentralized Smoothing ADMM for Quantile Regression with Non-Convex Sparse Penalties](https://mlsanity.com/paper/2408.01307/) (2024)
- [Distributed Semi-Supervised Sparse Statistical Inference](https://mlsanity.com/paper/2306.10395/) (2023)
- [High-Dimensional Differentially Private Quantile Regression: Distributed Estimation and Statistical Inference](https://mlsanity.com/paper/2508.05212/) (2025)
- [Divide-and-conquer methods for big data analysis](https://mlsanity.com/paper/2102.10771/) (2021)
- [Distributed Estimation and Inference for Semi-parametric Binary Response Models](https://mlsanity.com/paper/2210.08393/) (2022)
- [A Selective Review on Statistical Methods for Massive Data Computation: Distributed Computing, Subsampling, and Minibatch Techniques](https://mlsanity.com/paper/2403.11163/) (2024)
- [Using ARIMA to Predict the Expansion of Subscriber Data Consumption](https://mlsanity.com/paper/2404.15095/) (2024)
- [Federated Smoothing Proximal Gradient for Quantile Regression with Non-Convex Penalties](https://mlsanity.com/paper/2408.05640/) (2024)
- [Central Limit Theorems for Stochastic Gradient Descent Quantile Estimators](https://mlsanity.com/paper/2503.02178/) (2025)
- [Efficient Distributed Learning over Decentralized Networks with Convoluted Support Vector Machine](https://mlsanity.com/paper/2503.07563/) (2025)

## Abstract (arXiv)

This paper studies the inference problem in quantile regression (QR) for a large sample size n but under a limited memory constraint, where the memory can only store a small batch of data of size m. A natural method is the na"ive divide-and-conquer approach, which splits data into batches of size m, computes the local QR estimator for each batch, and then aggregates the estimators via averaging. However, this method only works when n=o(m2) and is computationally expensive. This paper proposes a computationally efficient method, which only requires an initial QR estimator on a small batch of data and then successively refines the estimator via multiple rounds of aggregations. Theoretically, as long as n grows polynomially in m, we establish the asymptotic normality for the obtained estimator and show that our estimator with only a few rounds of aggregations achieves the same efficiency as the QR estimator computed on all the data. Moreover, our result allows the case that the dimensionality p goes to infinity. The proposed method can also be applied to address the QR problem under distributed computing environment (e.g., in a large-scale sensor network) or for real-time streaming data.