A scalable projective scaling algorithm for l(p) loss with convex penalizations.

Area of Science:

• Optimization Algorithms
• Machine Learning Theory
• Convex Analysis

Background:

• Minimizing convex functions with lp loss is crucial in machine learning.
• Existing algorithms often lack explicit linear scalability with problem size.
• There's a need for efficient and scalable optimization methods in large-scale data analysis.

Purpose of the Study:

• To develop an accurate, efficient, and linearly scalable algorithm for minimizing convex functions with an lp loss component.
• To address the limitations of existing methods in terms of scalability for large datasets.
• To provide a robust optimization framework applicable to various machine learning problems.

Main Methods:

• Developed a second-order learning procedure with iterative descent for general convex penalization functions.
• Constructed efficient algorithms for functions satisfying Karmarkar's projective scaling condition.
• Introduced a lightweight, scalable message passing algorithm (MPA) through equivalent problem construction.

Main Results:

• The proposed MPA is intrinsically scalable, relying on matrix-vector multiplication and avoiding matrix inversion.
• MPA demonstrates global convergence for convex formulations and convergence to a stationary point for nonconvex cases.
• Extensive experiments validated the accuracy, efficiency, scalability, and applicability on sparse signal recovery, face classification, and dictionary learning.

Conclusions:

• The developed MPA offers a significant advancement in scalable optimization for machine learning.
• The algorithm's efficiency and scalability make it suitable for large-scale, real-world applications.
• The method's proven convergence properties ensure reliable performance across different problem types.