An ADMM Algorithm for a Class of Total Variation Regularized Estimation Problems (1203.1828v1)

Abstract: We present an alternating augmented Lagrangian method for convex optimization problems where the cost function is the sum of two terms, one that is separable in the variable blocks, and a second that is separable in the difference between consecutive variable blocks. Examples of such problems include Fused Lasso estimation, total variation denoising, and multi-period portfolio optimization with transaction costs. In each iteration of our method, the first step involves separately optimizing over each variable block, which can be carried out in parallel. The second step is not separable in the variables, but can be carried out very efficiently. We apply the algorithm to segmentation of data based on changes inmean (l_1 mean filtering) or changes in variance (l_1 variance filtering). In a numerical example, we show that our implementation is around 10000 times faster compared with the generic optimization solver SDPT3.

• The paper proposes a scalable ADMM algorithm specifically designed to exploit the structure of a class of total variation regularized estimation problems.
• The ADMM approach breaks down complex optimization problems into smaller, parallel subproblems, enabling impressive computational acceleration, particularly for large-scale data.
• Numerical results show the algorithm is significantly faster than existing solvers (e.g., 10000x for L1 mean filtering) and applicable to areas like signal processing and time-series analysis.

An ADMM Algorithm for Total Variation Regularized Estimation Problems

The paper explores an optimization technique using an Alternating Direction Method of Multipliers (ADMM) algorithm for solving a category of Total Variation (TV) regularized estimation problems. The research delves deeply into convex optimization problems characterized by an objective function defined as the sum of two terms. One term exhibits separability in variable blocks, while the other term is separable in the differences between consecutive variable blocks. The authors present several examples of such problems including Fused Lasso estimation, total variation denoising, and multi-period portfolio optimization.

Contributions and Methodology

The primary contribution of the paper is centered around developing a scalable and efficient algorithm tailored to exploit the specific structure of this subclass of optimization problems. The ADMM-based method allows simultaneous separable optimization over each block variable which enhances computational feasibility, particularly for large-scale problems. The implementation applies ADMM to $\ell_1$ mean filtering and $\ell_1$ variance filtering—both essential operations in signal processing, underpinning applications ranging from biological data analysis to financial data preprocessing.

The paper offers a comprehensive exposition of ADMM, detailing its convergence properties and its effectiveness in handling large-scale and high-dimensional datasets. The algorithm is implemented in parallel, a significant advancement that allows for an impressive computational acceleration—demonstrated to be several orders of magnitude faster than existing generic solvers like SDPT3 and CVX.

Key Numerical Results and Applications

The authors present a pronounced numerical performance, particularly with $\ell_1$ mean filtering for denoising applications, where their customized ADMM implementation outperforms traditional solvers by approximately a factor of 10000 in speed. Such enhancements are attributed to the algorithm's ability to break down the complex optimization into smaller, more manageable subproblems handled in parallel. Notably, the paper discusses parallel execution of specific ADMM steps, further enhancing efficiency, which can lead to additional speedups when executed on multicore architectures.

Applications of the proposed algorithm extend to domains requiring data segmentation and uncertainty estimation, with emphasis on the importance of piecewise constant estimates in scenarios like non-stationary data analysis. The fusion of mean and variance filtering complements the methodology's robustness, allowing for comprehensive statistical analysis in time-series datasets.

Practical and Theoretical Implications

Practical implications of this research lie in broadening the range of feasible applications of TV regularized estimation by reducing computational overhead. Theoretically, the research reinforces the utility of ADMM in decomposing composite optimization problems, providing insights into potential adaptations in other constrained settings.

Furthermore, the paper opens pathways for future research in the field of distributed optimization, emphasizing the exploration and tuning of optimization parameters such as the regularization parameter $\rho$. While the authors reference heuristic methods for parameter selection, further rigorous studies could elucidate optimizations that achieve convergence more rapidly or universally across different datasets.

In conclusion, the paper presents a methodologically sound and computationally efficient approach to handling TV regularized estimation problems, underscoring ADMM's capability in leveraging problem-specific structures to enhance performance in high-dimensional data environments. Further expansion and refinement of this framework promise significant advancements in various fields necessitating robust statistical estimation underpinned by optimization.