Embedded Feature Selection in Linear Support Vector Machines using a Scalable SDP Decomposition Approach

Core Concepts

This paper proposes novel mixed-integer formulations and scalable SDP-based relaxations to efficiently solve the embedded feature selection problem in linear Support Vector Machines, where a cardinality constraint is used to control the sparsity of the classifier.

Abstract

The paper studies the embedded feature selection problem in linear Support Vector Machines (SVMs), where a cardinality constraint is used to control the sparsity of the classifier. The problem is NP-hard due to the presence of the cardinality constraint.
The key contributions are:

Two novel mixed-integer formulations are introduced, based on a big-M reformulation and a complementarity constraint.
Novel SDP-based relaxations are proposed that can be decomposed into smaller conic problems, making the approach scalable.
Heuristic and exact algorithms are developed that exploit the decomposed relaxations to efficiently solve the problem.
Extensive numerical experiments demonstrate the effectiveness of the proposed approaches compared to off-the-shelf solvers.

The authors first analyze two mixed-integer formulations of the problem, one using a big-M reformulation and one using a complementarity constraint. They then propose several SDP-based relaxations, including decomposed versions that exploit the sparsity pattern of the problems to obtain much smaller and more scalable relaxations.
The heuristic algorithm uses the information from the relaxations to generate good feasible solutions, while the exact algorithm solves a sequence of mixed-integer second-order cone optimization problems to obtain the optimal solution. Numerical results on benchmark datasets show the efficiency and effectiveness of the proposed approaches.

Stats

The optimal solution w* of the standard SVM problem (1) usually has all nonzero components, so sparsity must be forced.
The cardinality constraint ∥w∥0 ≤ B in the FS-SVM problem (2) renders it NP-hard, even though the original SVM is solvable in polynomial time.

Quotes

"Even though objective function and all other constraints are convex quadratic or linear, the cardinality constraint ∥w∥0 ≤ B is of combinatorial type, which renders (2) NP-hard."
"The main contributions of our work are: - We analyze two novel Mixed Integer Quadratic optimization Problem (MIQP) formulations for the FS-SVM problem (2), based on tackling the ℓ0-pseudonorm constraint either by use of the big-M reformulation or by use of a complementarity constraint."

Key Insights Distilled From

Feature selection in linear SVMs via hard cardinality constraint: a scalable SDP decomposition approach

by Immanuel Bom... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10099.pdf

Deeper Inquiries

How could the proposed approaches be extended to handle nonlinear SVMs or other types of classifiers beyond linear SVMs

To extend the proposed approaches to handle nonlinear SVMs or other types of classifiers beyond linear SVMs, we can explore kernel methods. Kernel methods allow us to implicitly map the input data into a higher-dimensional feature space, where a linear classifier can be applied. By incorporating kernel functions into the feature selection process, we can handle nonlinear relationships between features and the target variable.
One approach could be to use the kernel trick to transform the input data into a higher-dimensional space, where a linear SVM can be applied. The feature selection process can then be performed in this transformed space, allowing for the selection of nonlinear combinations of features. Additionally, other types of classifiers, such as decision trees or neural networks, can also benefit from feature selection techniques.
By incorporating kernel methods and adapting the feature selection algorithms to work in the transformed feature space, we can effectively handle nonlinear SVMs and other types of classifiers.

What are the theoretical guarantees, in terms of approximation ratios or optimality gaps, that can be provided for the heuristic and exact algorithms

Theoretical guarantees for the heuristic algorithm:
The heuristic algorithm provides an upper bound solution for the feature selection problem. The guarantees for the heuristic algorithm can be analyzed in terms of its performance compared to the optimal solution.

Approximation Ratio: The heuristic algorithm can be evaluated based on its approximation ratio, which measures how close the heuristic solution is to the optimal solution. A theoretical analysis can provide insights into the quality of the heuristic solution compared to the optimal solution.

Optimality Gap: The optimality gap of the heuristic algorithm quantifies the difference between the objective value of the heuristic solution and the optimal solution. A smaller optimality gap indicates a more effective heuristic algorithm.

Theoretical guarantees for the exact algorithm:
The exact algorithm aims to find the optimal solution to the feature selection problem. The theoretical guarantees for the exact algorithm can include:

Optimality: The exact algorithm guarantees to find the optimal solution to the feature selection problem within a certain computational complexity.

Convergence: The exact algorithm should converge to the optimal solution as the number of iterations or steps increases.

Complexity Analysis: Theoretical guarantees can include complexity analysis, such as time complexity and space complexity, to provide insights into the efficiency of the algorithm.

By analyzing the approximation ratios, optimality gaps, convergence properties, and complexity analysis of both the heuristic and exact algorithms, we can assess their performance and effectiveness in solving the feature selection problem.

Can the decomposition techniques used in the SDP-based relaxations be applied to other types of structured sparsity-inducing constraints beyond the cardinality constraint considered here

The decomposition techniques used in the SDP-based relaxations can be applied to other types of structured sparsity-inducing constraints beyond the cardinality constraint considered in the context.

Group Sparsity: The decomposition techniques can be extended to handle group sparsity constraints, where features are grouped together, and the sparsity constraint applies to the groups rather than individual features. By decomposing the optimization problem based on the group structure, similar decomposition strategies can be applied to solve the problem efficiently.

Low-Rank Constraints: For problems involving low-rank constraints, the decomposition techniques can be utilized to decompose the optimization problem into smaller subproblems based on the low-rank structure. This can help in solving large-scale optimization problems with low-rank constraints efficiently.

Structured Sparsity: In cases where the sparsity pattern follows a structured pattern, such as block-sparsity or tree-sparsity, the decomposition techniques can be adapted to exploit the structured sparsity. By decomposing the problem based on the structured sparsity pattern, the optimization problem can be solved in a more scalable and efficient manner.

By applying the decomposition techniques to various types of structured sparsity-inducing constraints, we can enhance the scalability and efficiency of optimization algorithms for feature selection and other machine learning tasks.