Fast Alternating Linearization Methods for Minimizing the Sum of Two Convex Functions (0912.4571v2)

Abstract: We present in this paper first-order alternating linearization algorithms based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most $O(1/\epsilon)$ iterations to obtain an $\epsilon$-optimal solution, while our accelerated (i.e., fast) versions of them require at most $O(1/\sqrt{\epsilon})$ iterations, with little change in the computational effort required at each iteration. For both types of methods, we present one algorithm that requires both functions to be smooth with Lipschitz continuous gradients and one algorithm that needs only one of the functions to be so. Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones proposed by Goldfarb and Ma in [21] where the algorithms are Jacobi type methods. Numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms.

• The paper introduces first-order alternating linearization algorithms that achieve optimal iteration complexity of O(1/√ε) through accelerated methods.
• The proposed methods flexibly handle non-smooth regularizers, making them ideal for applications like compressed sensing, RPCA, and sparse inverse covariance selection.
• Empirical results validate the theoretical guarantees and practical efficiency of these techniques, setting a benchmark for scalable convex optimization research.

Overview of Fast Alternating Linearization Methods for Convex Optimization

The paper "Fast Alternating Linearization Methods for Minimizing the Sum of Two Convex Functions" by Donald Goldfarb, Shiqian Ma, and Katya Scheinberg presents a detailed paper of first-order alternating linearization algorithms based on an alternating direction augmented Lagrangian approach. These algorithms are proposed for minimizing functions that can be expressed as the sum of two convex components, a problem often encountered in the field of convex optimization. This class of problems is pertinent due to its applicability in various domains such as signal processing, machine learning, and statistical modeling.

Algorithms and Key Contributions

The paper introduces two variants of alternating linearization methods: basic and accelerated versions. The basic methods require $O(1/\epsilon)$ iterations to achieve an $\epsilon$-optimal solution, while the accelerated (or "fast") versions require $O(1/\sqrt{\epsilon})$ iterations, enhancing efficiency significantly with minimal additional computational cost per iteration. The research highlights the Gauss-Seidel type methods that alternate between linearization with respect to each component function in contrast to Jacobi type methods used previously by Goldfarb and Ma.

Key contributions of this paper include:

• Complexity Bounds: The authors provide rigorous iteration complexity bounds for both types of methods. Their accelerated algorithms achieve the theoretical optimal complexity for first-order methods given by $O(\sqrt{L/\epsilon})$, where $L$ is the Lipschitz constant of the gradient of the smooth component of the objective function.
• Flexibility with Non-Smooth Functions: The algorithms handle cases where only one of the functions is smooth, which is crucial for applications involving non-smooth regularization terms like the $\ell_1$ norm often used for sparse solutions.
• Practical Algorithms: Empirical results support the theoretical findings, indicating that the proposed methods are not only theoretically sound but also practically effective for large-scale problems.

Numerical Results and Applications

Applications illustrated in this paper are diverse, covering:

1. Compressed Sensing ($\ell_1$ Minimization): The paper discusses the application of the methods to restore sparse signals from a reduced set of measurements, a central problem in compressed sensing.
2. Robust Principal Component Analysis (RPCA): The RPCA formulation accommodates separable constraints that are suited for uncovering low-rank and sparse matrices, demonstrating the methods' utility in computer vision and image processing.
3. Sparse Inverse Covariance Selection (SICS): The methods are shown to effectively select a sparse inverse covariance in Gaussian graphical models, which are crucial for discovering structures within high-dimensional statistical data.
4. Matrix Completion with Missing Data: When data matrices are corrupted or incomplete, these methods demonstrate robustness in recovering the underlying matrix structures, as seen in both synthetic and real-world datasets.

Speculation on Future Work

This research opens avenues for future investigation into extending alternating linearization methods to broader classes of non-linear optimization problems that incorporate constraints beyond simple addition of convex functions. Enhancements to handle dynamic optimization problems and those involving time-varying data using these techniques could significantly benefit real-time applications. Moreover, integrating machine learning frameworks for adaptive parameter tuning within these algorithms could lead to more autonomous and efficient optimization procedures.

In summary, the paper presents significant advancements in the development of alternating linearization methods for convex optimization problems. It provides strong theoretical guarantees and empirical validations, making it a solid benchmark for subsequent research in this domain.