Conditional Gradient Algorithms for Norm-Regularized Smooth Convex Optimization (1302.2325v4)

Abstract: Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $|\cdot|$ and a smooth convex function $f$, we want either 1) to minimize the norm over the intersection of the cone and a level set of $f$, or 2) to minimize over the cone the sum of $f$ and a multiple of the norm. We focus on the case where (a) the dimension of the problem is too large to allow for interior point algorithms, (b) $|\cdot|$ is "too complicated" to allow for computationally cheap Bregman projections required in the first-order proximal gradient algorithms. On the other hand, we assume that {it is relatively easy to minimize linear forms over the intersection of $K$ and the unit $|\cdot|$-ball}. Motivating examples are given by the nuclear norm with $K$ being the entire space of matrices, or the positive semidefinite cone in the space of symmetric matrices, and the Total Variation norm on the space of 2D images. We discuss versions of the Conditional Gradient algorithm capable to handle our problems of interest, provide the related theoretical efficiency estimates and outline some applications.

• The paper introduces novel conditional gradient variants that efficiently solve norm-regularized convex optimization problems in high-dimensional settings.
• It provides theoretical efficiency estimates demonstrating improved convergence performance over standard proximal methods in complex norm contexts.
• The study validates these methods with practical applications such as matrix completion and TV-norm image reconstruction, supported by strong numerical results.

Conditional Gradient Algorithms for Norm-Regularized Smooth Convex Optimization: A Review

In their research paper, Zaid Harchaoui, Anatoli Juditsky, and Arkadi Nemirovski explore conditional gradient algorithms tailored for solving two distinct types of convex optimization problems that are frequently encountered in signal processing and machine learning. The two primary formulations addressed in this paper are: 1) minimizing the norm over the intersection of a cone and a level set of a smooth convex function, and 2) minimizing the sum of a smooth convex function and a norm regularization term over a cone. The authors focus on scenarios where the dimensionality of problems precludes the application of traditional interior-point methods, and where the complexity of the chosen norm impedes efficient computation via Bregman projections commonly utilized in first-order proximal methods.

Their investigation is motivated by practical applications involving complex norms such as the nuclear norm in matrix spaces and the Total Variation norm in image processing. To address these challenges, the paper advances the Conditional Gradient (CG) algorithm—also known as the Frank-Wolfe algorithm—emphasizing its applicability to large-scale optimization problems without requiring expensive sub-problem solutions characteristic of proximal methods.

Key Contributions

1. Algorithmic Development: The paper develops variants of conditional gradient algorithms suited for the optimization problems under consideration, providing a detailed methodological framework that captures the nuances of applying CG methods in a computationally efficient manner.
2. Efficiency Analysis: The authors provide theoretical efficiency estimates for these algorithms, substantiating their applicability in high-dimensional settings while ensuring computational feasibility.
3. Practical Applications: By illustrating the use of CG algorithms in contexts such as matrix completion and TV-norm-based image reconstruction, the paper highlights the practical utility and versatility of these methods within real-world data-intensive applications.
4. Conditional Gradient with Memory: The authors extend the conventional CG algorithm to incorporate a memory-based approach, allowing for more refined updates by retaining information from previous iterations. This variant, labeled CG with memory (CGM), demonstrates improved performance by leveraging past computed solutions to inform current decisions, thus accelerating convergence.

Numerical Results and Implications

The paper presents strong numerical results showing instances where CG algorithms outperform other first-order methods, particularly in scenarios involving complex norm structures or when only linear minimization over feasible sets is computationally tenable. Moreover, the CGM variant shows promising computational efficiency, often attaining superior results with reduced memory and time complexity compared to standard CG.

Theoretical and Practical Implications

The theoretical advancements proposed in this research have profound implications for both the theory and practice of large-scale convex optimization. The CG methodologies not only broaden the scope of feasible applications in high-dimensional spaces but also contribute to narrowing the computational gap where traditional methods fall short.

This paper lays a compelling foundation for future research paths aiming to further refine CG algorithms, potentially integrating adaptive schemes or stochastic variants that might enhance their applicability to a wider range of optimization problems. The robust theoretical underpinning and demonstrated efficacy across various settings ensure that CG algorithms remain a relevant and potent tool in the arsenal of optimization techniques for machine learning and computational statistics.

In summary, this paper is a significant contribution to the field of optimization, providing both theoretical innovation and practical algorithms for addressing complex norm-regularized convex optimization problems efficiently.