[Paper Review] Efficient numerical algorithms for regularized regression problem with applications to traffic matrix estimations

Anikin, Anton, Dvurechensky, Pavel+6 more|arXiv (Cornell University)|Aug 4, 2015

Sparse and Compressive Sensing Techniques18 references3 citations

TL;DR

This paper proposes and compares efficient numerical algorithms for solving large-scale regularized regression problems with sparse 0-1 constraint matrices, particularly applied to traffic matrix estimation in IP networks. By exploiting the special structure of the constraints, the authors demonstrate that specialized solvers significantly outperform general-purpose methods in convergence speed and scalability.

ABSTRACT

In this work we collect and compare to each other many different numerical methods for regularized regression problem and for the problem of projection on a hyperplane. Such problems arise, for example, as a subproblem of demand matrix estimation in IP- networks. In this special case matrix of affine constraints has special structure: all elements are 0 or 1 and this matrix is sparse enough. We have to deal with huge-scale convex optimization problem of special type. Using the properties of the problem we try "to look inside the black-box" and to see how the best modern methods work being applied to this problem.

Motivation & Objective

• Address the challenge of estimating demand matrices in IP networks using regularized regression with affine constraints.
• Tackle large-scale convex optimization problems arising in traffic matrix estimation, where constraint matrices are sparse and binary.
• Evaluate and compare modern numerical methods tailored for this specific problem structure to identify optimal solvers.
• Improve computational efficiency and scalability for real-world network monitoring applications.
• Investigate the performance of optimization algorithms when applied to structured, sparse constraint matrices in practice.

Proposed method

• Formulate the traffic matrix estimation problem as a regularized regression with affine equality constraints.
• Utilize the sparsity and binary (0-1) structure of the constraint matrix to design problem-specific solvers.
• Apply and compare multiple numerical methods, including first-order methods like FISTA, Nesterov's optimal method, and adaptive restart techniques.
• Implement and test projection algorithms onto sparse hyperplanes to accelerate convergence.
• Optimize algorithmic parameters based on problem-specific properties to enhance computational efficiency.
• Use a combination of theoretical analysis and empirical benchmarking to evaluate performance on real-world network data.

Experimental results

Research questions

• RQ1 Which numerical algorithms achieve the fastest convergence for large-scale regularized regression with sparse 0-1 constraint matrices?
• RQ2 How does exploiting the structural sparsity and binary nature of the constraint matrix improve solver performance?
• RQ3 What is the relative efficiency of first-order methods versus second-order or interior-point methods in this context?
• RQ4 Can adaptive restart strategies significantly accelerate convergence in this class of problems?
• RQ5 How do the proposed solvers scale with increasing problem size in real-world traffic matrix estimation?

Key findings

• First-order methods such as FISTA with adaptive restart outperform general-purpose solvers in convergence speed and scalability.
• Leveraging the sparse 0-1 structure of the constraint matrix enables significant computational gains over generic optimization approaches.
• The proposed problem-specific solvers achieve faster convergence and lower computational cost on large-scale network data.
• Adaptive restart techniques improve the practical performance of Nesterov-type methods in the context of traffic matrix estimation.
• Projection onto sparse hyperplanes is computationally efficient and enables faster iteration in the optimization loop.
• Empirical results show that specialized solvers reduce solution time by orders of magnitude compared to standard methods on large-scale instances.