Optimization with First-Order Surrogate Functions (1305.3120v1)

Abstract: In this paper, we study optimization methods consisting of iteratively minimizing surrogates of an objective function. By proposing several algorithmic variants and simple convergence analyses, we make two main contributions. First, we provide a unified viewpoint for several first-order optimization techniques such as accelerated proximal gradient, block coordinate descent, or Frank-Wolfe algorithms. Second, we introduce a new incremental scheme that experimentally matches or outperforms state-of-the-art solvers for large-scale optimization problems typically arising in machine learning.

• The paper unifies various first-order methods under a majorization-minimization framework, linking approaches like accelerated proximal gradient and Frank-Wolfe.
• It introduces the MISO algorithm, an incremental scheme that achieves linear convergence for strongly convex objectives in large-scale optimization.
• Empirical results validate MISO’s efficiency in regularized logistic regression, highlighting its potential for high-dimensional machine learning applications.

Optimization with First-Order Surrogate Functions

The paper "Optimization with First-Order Surrogate Functions" by Julien Mairal offers a comprehensive exploration of optimization techniques through the lens of using surrogate functions. This framework provides a unified perspective on a variety of first-order optimization methods, enhancing both theoretical understanding and practical algorithmic implementations.

Main Contributions

The research makes two pivotal contributions to the field of optimization:

1. Unified Framework for First-Order Methods: The paper provides a cohesive view of several first-order optimization algorithms, effectively bridging methodologies such as accelerated proximal gradient, block coordinate descent, and Frank-Wolfe algorithms. By conceptualizing these techniques under the umbrella of majorization-minimization, it unifies them through the iterative minimization of surrogate functions.
2. Development of the MISO Algorithm: Introduced is the MISO (Minimization by Incremental Surrogate Optimization) algorithm, which serves as an incremental scheme potentially surpassing state-of-the-art solvers, especially for large-scale optimization tasks prevalent in machine learning. MISO is notably positioned alongside methods like SAG and SDCA in terms of demonstrating linear convergence rates for strongly convex objectives.

Theoretical Framework

The paper underlines the versatility of majorization-minimization principles, extending its relevance from classical applications in signal processing and machine learning to new algorithmic strategies. The first-order surrogate functions, pivotal to this discourse, approximate non-smooth objectives up to a smooth error term, allowing the convergence analysis to predict stationary points for non-convex problems and establish explicit rates for convex scenarios.

Numerical Results and Claims

One of the striking claims supported by experimental results is that MISO matches or exceeds the performance of contemporary solvers in implementing large-scale regularized logistic regression. This empirical validation spans over datasets inheriting varied dimensions and complexities.

Implications and Future Work

The implications of this research are multifaceted:

• Empirical Algorithms:

It promises enhanced performance in optimization problems implicated with high-dimensional data, supportive through empirical evidence. This has direct applications in machine learning models, especially when handling extensive datasets efficiently.

• Theoretical Contributions:

By providing a broad-spectrum analysis of convergence and rate proofs across different problem sets, it serves as a foundational work for further exploration into optimization with advanced surrogate mechanisms.

Future directions mentioned include investigating fully stochastic variants of the presented framework, which could optimize the practicality of the methodology for handling even larger datasets, mitigating memory constraints along the way.

In conclusion, the paper robustly contributes to both the theoretical development and experimental prowess of optimization methodologies, leveraging first-order surrogates to enhance existing algorithms and propose new ones within the machine learning sector. With both the promise of practical efficiencies and theoretical advancements, the framework lays down a robust antecedent for further research continuation in this domain.