Sampling Algorithms and Coresets for Lp Regression (0707.1714v1)

Abstract: The Lp regression problem takes as input a matrix $A \in \Real^{n \times d}$, a vector $b \in \Real^n$, and a number $p \in [1,\infty)$, and it returns as output a number ${\cal Z}$ and a vector $x_{opt} \in \Real^d$ such that ${\cal Z} = \min_{x \in \Real^d} ||Ax -b||p = ||Ax{opt}-b||_p$. In this paper, we construct coresets and obtain an efficient two-stage sampling-based approximation algorithm for the very overconstrained ($n \gg d$) version of this classical problem, for all $p \in [1, \infty)$. The first stage of our algorithm non-uniformly samples $\hat{r}_1 = O(36^p d^{\max{p/2+1, p}+1})$ rows of $A$ and the corresponding elements of $b$, and then it solves the Lp regression problem on the sample; we prove this is an 8-approximation. The second stage of our algorithm uses the output of the first stage to resample $\hat{r}_1/\epsilon^2$ constraints, and then it solves the Lp regression problem on the new sample; we prove this is a $(1+\epsilon)$-approximation. Our algorithm unifies, improves upon, and extends the existing algorithms for special cases of Lp regression, namely $p = 1,2$. In course of proving our result, we develop two concepts--well-conditioned bases and subspace-preserving sampling--that are of independent interest.

Sampling Algorithms and Coresets for $\ell_p$ Regression

The paper presents a comprehensive paper of sampling algorithms and the construction of coresets for the $\ell_p$ regression problem, which is defined as finding a vector $x$ that minimizes the $p$-norm of the residuals between the intended solutions $Ax$ and a given target vector $b$. This problem is widely relevant, as it underpins applications in statistical data analysis and machine learning, particularly in dealing with overconstrained systems where the number of conditions $n$ greatly exceeds the number of variables $d$.

Key Contributions

1. Two-Stage Sampling Algorithm: The authors introduce an efficient two-stage sampling-based approximation algorithm for the overconstrained $\ell_p$ regression problem, providing a $(1+\epsilon)$-approximation solution. The primary highlight of this method is its application across all $p \in [1, \infty)$, which generalizes previous approaches tailored for specific values of $p$ such as $p=1$ and $p=2$.
2. Development of New Concepts:
   • Well-Conditioned Bases: These are introduced as a novel tool for capturing the geometry of $\ell_p$ norms and are instrumental in the sampling process. A well-conditioned basis ensures that for any vector $z$, the $q$-norm is a close approximation to the $p$-norm space in consideration.
   • Subspace-Preserving Sampling: This method minimizes sampling variance, preserving the relevant subspace information needed for a reliable approximation of the original regression problem.
3. Enhanced Sampling Techniques: Through an innovative interplay between initial coarse approximations and refined second-stage sampling based on calculated residuals, the method not only achieves significant improvements in computational efficiency but also preserves essential structural information of the problem space.

Numerical and Theoretical Implications

• Numerical Efficiency: The two-stage sampling strategy significantly reduces the number of rows (samples) needed to achieve a $(1+\epsilon)$-approximation. For less computational complexity, it leverages an optimal combination of matrix and residual information for sampling, in contrast to relying solely on matrix characteristics or solving the entire regression problem upfront.
• Implications for High-Dimensional Data: By constructing coresets that are polynomial in dimension and thus applicable to high-dimensional data, the paper bridges an important gap, allowing algorithms previously applied primarily to low-dimensional spaces now to be efficient in higher dimensions.

Future Directions in AI

Given the broad applicability of overconstrained regression problems in AI and data science, the presented methods have potential far-reaching impacts:

• Scalability to Large Datasets: The reduction in computational requirements will facilitate the application of $\ell_p$ regression to extremely large datasets, which is increasingly common in AI-driven tasks such as natural language processing and computer vision.
• Real-Time Processing: Faster approximation algorithms could empower real-time decision-making systems that utilize regression models to handle dynamic data inputs efficiently.
• Generalization to Other Norms: Further exploration could emerge by extending these concepts to other norms beyond the conventional $\ell_p$, potentially opening new possibilities for optimization in unsupervised learning tasks.

In summary, the paper provides an in-depth mathematical framework and algorithmic strategy for solving the $\ell_p$ regression problem with improved efficiency and applicability. Its contributions to the field underscore essential mechanisms for operating within high-dimensional spaces and overconstrained conditions, which are prevalent across modern data-driven disciplines.