Efficient Optimization of Strongly Convex Functions with Linear Constraints using Accelerated Randomized Bregman-Kaczmarz Method

Core Concepts

The authors propose an accelerated randomized Bregman-Kaczmarz method to efficiently solve linearly constrained optimization problems with strongly convex (possibly non-smooth) objective functions. They provide theoretical analysis showing linear convergence rates and demonstrate the superior efficiency of the proposed method compared to existing approaches.

Abstract

The paper considers the problem of approximating solutions of large-scale consistent linear systems Ax = b, where the full matrix A is not accessible simultaneously. The authors aim to find the unique solution characterized by a general strongly convex function f(x), subject to the linear constraints Ax = b. The key highlights and insights are: The authors propose a block (accelerated) randomized Bregman-Kaczmarz method that only uses a block of constraints in each iteration to tackle this problem. They consider a dual formulation of the problem to deal efficiently with the linear constraints. Using convex tools, they show that the dual function satisfies the Polyak-Lojasiewicz (PL) property, provided that the primal objective function is strongly convex and satisfies some mild assumptions. The authors transfer the algorithm to the primal space, which combined with the PL property, allows them to obtain linear convergence rates for their proposed method. The authors provide a theoretical analysis of the convergence of their proposed method under different assumptions on the objective function and demonstrate its superior efficiency and speed compared to existing methods for the same problem. The authors also propose a restart scheme for their accelerated method that exhibits faster convergence than the standard counterpart.

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Acceleration and restart for the randomized Bregman-Kaczmarz method

by Lionel Tondj... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2310.17338.pdf

Deeper Inquiries

How can the proposed method be extended to handle non-convex objective functions or more general constraints beyond linear equalities

To extend the proposed method to handle non-convex objective functions or more general constraints beyond linear equalities, several modifications and adaptations can be made. One approach could involve incorporating regularization terms or penalty functions to handle non-convexity. By introducing additional terms that promote sparsity or structure in the solution, the algorithm can be adapted to handle non-convex objectives. Additionally, techniques from non-convex optimization, such as stochastic gradient descent or metaheuristic algorithms, can be integrated into the method to explore a wider range of objective functions and constraints. Furthermore, the use of advanced optimization techniques like proximal operators or alternating minimization can help address non-convexity and more complex constraints in the optimization problem.

What are the potential applications of the accelerated Bregman-Kaczmarz method beyond the sparse optimization problem considered in the paper

The accelerated Bregman-Kaczmarz method has potential applications beyond the sparse optimization problem discussed in the paper. Some potential applications include: Image Processing: The method can be applied to image reconstruction problems, such as MRI or CT image reconstruction, where the objective is to recover high-quality images from undersampled data. Signal Processing: In signal processing tasks like audio signal denoising or source separation, the method can be used to efficiently estimate the underlying signals from noisy observations. Machine Learning: The method can be utilized in training deep learning models, particularly in scenarios where data is distributed across multiple locations and communication constraints exist. Data Science: For large-scale data analysis tasks like matrix completion or collaborative filtering, the method can help in efficiently handling missing data and making accurate predictions.

Can the theoretical analysis be further improved to obtain tighter convergence rates or to relax the assumptions on the objective function

To improve the theoretical analysis and obtain tighter convergence rates or relax the assumptions on the objective function, several strategies can be considered: Advanced Regularization Techniques: Incorporating advanced regularization techniques like adaptive regularization or non-convex penalties can help in handling a broader class of objective functions. Robust Optimization: Introducing robust optimization methods to account for uncertainties or noise in the data can lead to more robust convergence guarantees. Distributed Optimization: Extending the analysis to distributed optimization settings can provide insights into the convergence behavior under communication constraints and decentralized computation. Non-Smooth Optimization: Considering non-smooth optimization techniques like subgradient methods or proximal operators can help relax the smoothness assumptions on the objective function while still ensuring convergence properties. Empirical Validation: Conducting extensive numerical experiments on a diverse set of problems can validate the theoretical results and provide insights into the practical performance of the method under various scenarios.