[Paper Review] Gradient-based Sampling: An Adaptive Importance Sampling for Least-squares

Rong Zhu|arXiv (Cornell University)|Mar 2, 2018

Sparse and Compressive Sensing Techniques22 citations

TL;DR

This paper proposes gradient-based sampling, an adaptive importance sampling method for least-squares problems that uses both input data and response variables to compute sampling probabilities based on gradient values. By reducing computation time to O(nd) and achieving tighter error bounds, it outperforms leverage-based and uniform sampling in both statistical efficiency and computational speed on synthetic and real datasets.

ABSTRACT

In modern data analysis, random sampling is an efficient and widely-used strategy to overcome the computational difficulties brought by large sample size. In previous studies, researchers conducted random sampling which is according to the input data but independent on the response variable, however the response variable may also be informative for sampling. In this paper we propose an adaptive sampling called the gradient-based sampling which is dependent on both the input data and the output for fast solving of least-square (LS) problems. We draw the data points by random sampling from the full data according to their gradient values. This sampling is computationally saving, since the running time of computing the sampling probabilities is reduced to O(nd) where n is the full sample size and d is the dimension of the input. Theoretically, we establish an error bound analysis of the general importance sampling with respect to LS solution from full data. The result establishes an improved performance of the use of our gradient- based sampling. Synthetic and real data sets are used to empirically argue that the gradient-based sampling has an obvious advantage over existing sampling methods from two aspects of statistical efficiency and computational saving.

Motivation & Objective

• To address the computational burden of large-scale least-squares problems in modern data analysis.
• To overcome the limitations of uniform and leverage-based sampling, which ignore response variable information and incur high computational costs.
• To develop a sampling method that leverages both input and output data to improve statistical efficiency and reduce computation time.
• To establish a non-asymptotic error bound for general importance sampling, enabling theoretical performance guarantees for the proposed method.

Proposed method

• The method computes sampling probabilities using the gradient of the loss function with respect to the parameters, based on a pilot estimate of the least-squares solution.
• Sampling probabilities are proportional to the magnitude of the gradient vector for each data point, making them responsive to both input features and response values.
• The algorithm reduces the time to compute sampling probabilities to O(nd), significantly faster than leverage-based methods requiring SVD or randomized SVD.
• It uses importance sampling with non-uniform probabilities derived from gradient values to form a subsampled least-squares problem.
• Theoretical analysis derives a non-asymptotic error bound for the LS solution under general importance sampling, showing improved performance guarantees.
• The method is extended to real-world and synthetic datasets to empirically validate its statistical and computational advantages.

Experimental results

Research questions

• RQ1 Can incorporating response variable information into sampling probabilities improve statistical efficiency in least-squares estimation?
• RQ2 Does gradient-based sampling reduce computational cost compared to leverage-based sampling while maintaining or improving solution accuracy?
• RQ3 Can a non-asymptotic error bound be established for importance sampling in least-squares problems, and how does it compare to existing bounds?
• RQ4 How does the performance of gradient-based sampling vary across different data distributions and sample sizes in practice?
• RQ5 Can the proposed method achieve near-minimum error bounds in the least-squares solution, as theoretically predicted?

Key findings

• Gradient-based sampling reduces the time to compute sampling probabilities to O(nd), offering a significant computational advantage over leverage-based sampling.
• Theoretical analysis establishes a non-asymptotic error bound for importance sampling that shows gradient-based sampling can nearly achieve the minimum possible error bound.
• Empirical results on synthetic and real datasets show that gradient-based sampling achieves higher statistical efficiency than leverage-based and uniform sampling.
• The method demonstrates improved performance in balancing subsample size and statistical accuracy, as supported by the non-asymptotic error bound.
• When the pilot estimate is close to the true solution, the sampling variance of the gradient-based method approaches the theoretical minimum, as shown by σ²_b(π⁰) − σ²_b(πᵉ) = o_p(1).
• The method achieves a high-probability error bound of the form Pr{||β̃ − β|| ≤ C₁r⁻¹ᐟ² + C₂r⁻¹} ≥ 1 − δ, with C₁ and C₂ depending on the data and confidence level.