Incremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning (1402.4419v3)

Abstract: Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function. These upper bounds are tight at the current estimate, and each iteration monotonically drives the objective function downhill. Such a simple principle is widely applicable and has been very popular in various scientific fields, especially in signal processing and statistics. In this paper, we propose an incremental majorization-minimization scheme for minimizing a large sum of continuous functions, a problem of utmost importance in machine learning. We present convergence guarantees for non-convex and convex optimization when the upper bounds approximate the objective up to a smooth error; we call such upper bounds "first-order surrogate functions". More precisely, we study asymptotic stationary point guarantees for non-convex problems, and for convex ones, we provide convergence rates for the expected objective function value. We apply our scheme to composite optimization and obtain a new incremental proximal gradient algorithm with linear convergence rate for strongly convex functions. In our experiments, we show that our method is competitive with the state of the art for solving machine learning problems such as logistic regression when the number of training samples is large enough, and we demonstrate its usefulness for sparse estimation with non-convex penalties.

• The paper introduces an incremental majorization-minimization approach that optimizes large-scale ML models by updating surrogate functions for individual components.
• It establishes convergence guarantees for both convex and non-convex problems, including a linear rate for strongly convex cases.
• Empirical results on logistic regression and sparse estimation demonstrate that the method competes with state-of-the-art solvers in scalability and performance.

Incremental Majorization-Minimization Optimization for Large-Scale Machine Learning

This paper introduces a novel optimization technique termed "Incremental Majorization-Minimization Optimization" (IMM), designed to tackle large-scale machine learning problems. The principle of majorization-minimization (MM) involves minimizing a sequence of upper bounds of the objective function, wherein each iteration guarantees a decrease in the objective function's value. Such a framework is particularly appealing in fields like signal processing and statistics but often faces scalability issues in machine learning. This paper addresses these challenges by proposing an incremental method that maintains efficiency even with a large number of function components.

Methodological Contributions

The authors present a new scheme for optimizing a large sum of continuous functions by incrementally applying a majorization-minimization approach. The key contributions include:

1. Incremental Framework: The proposed scheme systematically updates surrogate functions for each individual function component rather than the whole objective function, thus reducing computation per iteration.
2. Convergence Guarantees: For both convex and non-convex optimization problems, convergence to asymptotic stationary points is established. For convex settings, the paper provides convergence rates, including a linear rate for strongly convex functions.
3. Algorithm Variants: The paper introduces different variants of the MISO (Minimization by Incremental Surrogate Optimization) approach, suitable for specific types of machine learning problems like logistic regression and sparse estimation. These variants illustrate the algorithm's versatility across different problem structures.
4. Numerical Experiments: The paper reports competitive empirical results showcasing that the method performs on par with existing state-of-the-art solvers for large-scale machine learning tasks, notably logistic regression, underlining the practical efficacy of the approach.

Implications and Future Directions

The research provides a robust framework that addresses scalability in machine learning optimization, which is critical given the increasing size of datasets. The implications are twofold:

1. Practical Significance: The ability to handle large datasets with efficiency and convergence assurance is significant for machine learning practitioners. This is particularly relevant in scenarios where traditional methods might be computationally prohibitive.
2. Theoretical Foundations: From a theoretical standpoint, the work extends the understanding of MM algorithms and their convergence properties, providing a basis for further exploration of incremental optimization techniques.

Looking forward, there are intriguing avenues for further research. Firstly, the exploration of adaptive techniques for selecting the approximation parameters and step sizes can potentially enhance the performance of IMM algorithms. Secondly, investigating the extension of IMM approaches to other machine learning paradigms, such as deep learning, offers promising research directions. Lastly, given the effectiveness of IMM in non-convex settings, further theoretical exploration into its applications can open new frontiers in optimization research.

In conclusion, this paper makes significant strides in the optimization of large-scale machine learning models by effectively utilizing the majorization-minimization paradigm within an incremental framework. The blend of strong theoretical insights with practical utility makes it a noteworthy contribution to the field.