Anderson Acceleration for Iteratively Reweighted ℓ1 Algorithm: Convergence and Complexity Analysis

核心概念

The author proposes an Anderson-accelerated IRL1 algorithm, demonstrating local linear convergence without the need for Kurdyka-Lojasiewicz conditions, outperforming existing Nesterov acceleration-based algorithms.

要約

The content discusses the development of Anderson acceleration in optimization algorithms, specifically focusing on the proposed Anderson-accelerated IRL1 algorithm. It addresses challenges in convergence and complexity analysis, highlighting its theoretical results and experimental performance compared to existing methods.
The iteratively reweighted ℓ1 (IRL1) algorithm is explored for solving nonconvex and nonsmooth optimization problems with sparse regularization. The article delves into the application of Anderson acceleration to enhance fixed-point iteration speed and establish local linear convergence rates.
Key points include the introduction of Anderson acceleration for speeding up fixed-point iteration processes, overcoming challenges posed by nonconvex and nonsmooth optimization problems. The proposed AAIRL1 algorithm showcases improved performance over traditional Nesterov acceleration-based approaches.
Experimental results indicate that the new algorithm surpasses existing methods in terms of convergence rates and computational efficiency. The incorporation of a globally convergent strategy through nonmonotone line search conditions ensures robustness in achieving convergence across various optimization scenarios.

統計

Recently, Anderson acceleration has gained prominence owing to its exceptional performance for speeding up fixed-point iteration.
Experimental results indicate that the proposed algorithm outperforms existing Nesterov acceleration-based algorithms.
The algorithm has local linear convergence when the KL coefficient is not greater than 1/2.
Wang et al. introduced an accelerated iteratively reweighted L1 algorithm with Nesterov technique specifically for LPN regularization.
For a fixed point issue that can be divided into a smooth component and a non-smooth component with a minor Lipschitz constant, new findings have been demonstrated by Bian et al.

引用

抽出されたキーインサイト

Anderson acceleration for iteratively reweighted $\ell_1$ algorithm

by Kexin Li 場所 arxiv.org 03-13-2024

https://arxiv.org/pdf/2403.07271.pdf

深掘り質問

How does incorporating nonmonotone line search conditions impact the global convergence of optimization algorithms

Incorporating nonmonotone line search conditions in optimization algorithms can have a significant impact on global convergence. Nonmonotone line search allows for greater flexibility in accepting steps during the optimization process, as it relaxes the strict requirement of a decreasing function value at each iteration. This relaxation enables the algorithm to explore a wider range of solutions and potentially escape local minima more effectively. By incorporating nonmonotone line search, optimization algorithms can strike a balance between exploration and exploitation, leading to improved convergence properties.

What are potential limitations or drawbacks of using Anderson acceleration in complex optimization scenarios

While Anderson acceleration is a powerful technique for accelerating convergence in iterative methods, there are potential limitations and drawbacks when applied to complex optimization scenarios. One limitation is that Anderson acceleration may not guarantee global convergence in all cases, especially when dealing with highly nonlinear or non-convex functions. The method relies on historical iterates and weighted sums, which could introduce oscillations or instability under certain conditions. Additionally, determining suitable parameters such as the history length or weight coefficients can be challenging and may require manual tuning for optimal performance.

How can the principles behind Anderson acceleration be applied to other areas outside of mathematical optimization

The principles behind Anderson acceleration can be applied beyond mathematical optimization to various other areas where iterative methods are used. For example:

Signal Processing: Anderson acceleration techniques can enhance signal processing algorithms by speeding up convergence rates in tasks like image denoising or audio signal enhancement.

Machine Learning: In machine learning models that involve iterative processes like gradient descent or backpropagation, Anderson acceleration can improve training efficiency by accelerating convergence towards optimal solutions.

Physics Simulations: Computational physics simulations often rely on iterative solvers for complex systems of equations. Applying Anderson acceleration can reduce computational time in solving these systems accurately.

Data Science: Optimization problems in data science applications such as clustering algorithms or dimensionality reduction techniques could benefit from faster convergence using Anderson acceleration methods.

By leveraging the concepts of historical information utilization and weighted averaging from Anderson acceleration, advancements in various fields requiring iterative computations are possible with improved efficiency and speed of convergence.