Parallel Selective Algorithms for Big Data Optimization (1402.5521v5)

Abstract: We propose a decomposition framework for the parallel optimization of the sum of a differentiable (possibly nonconvex) function and a (block) separable nonsmooth, convex one. The latter term is usually employed to enforce structure in the solution, typically sparsity. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss- Seidel (i.e., sequential) ones, as well as virtually all possibilities "in between" with only a subset of variables updated at each iteration. Our theoretical convergence results improve on existing ones, and numerical results on LASSO, logistic regression, and some nonconvex quadratic problems show that the new method consistently outperforms existing algorithms.

• The paper introduces a flexible parallel selective decomposition framework for optimizing the sum of differentiable nonconvex and block-separable convex functions.
• The approach provides enhanced theoretical convergence results and empirically outperforms conventional algorithms on large-scale problems.
• Offering flexibility and practical application across diverse fields, this framework provides a powerful approach for big data optimization challenges.

Overview of Parallel Selective Algorithms for Nonconvex Big Data Optimization

The paper introduces a sophisticated decomposition framework designed for optimizing the sum of a differentiable, potentially nonconvex function, and a nonsmooth, convex, block-separable function. This combination is commonly encountered in numerous practical problems across diverse fields like machine learning and signal processing, where the nonsmooth term is often utilized to enforce sparsity in solutions.

The proposed framework is noted for its flexibility, encompassing a variety of parallel processing strategies that range from fully parallel (Jacobi schemes) to sequential (Gauss-Seidel schemes), as well as hybrid techniques. A critical contribution of this work is the enhancement of theoretical convergence results compared to existing methods. Additionally, empirical evidence suggests that this new approach consistently outperforms conventional algorithms in solving problems like the LASSO, logistic regression, and nonconvex quadratic programming, highlighting its practical efficacy.

Key Contributions

1. Decomposition Framework:
   • The framework optimizes a function composed of a differentiable nonconvex part and a block-separable convex part.
   • It incorporates mechanisms to update only a subset of variables in each iteration, which balances computation costs and convergence speed.
2. Theoretical Advancements:
   • The convergence proofs presented improve upon existing theoretical results, particularly for problems involving nonconvex functions.
   • The paper ensures that even when a subset of variables is updated, convergence to stationary solutions is guaranteed.
3. Methodological Flexibility:
   • The framework allows selecting various levels of parallelism tailored to the computational architecture.
   • Users can choose different approximation techniques for the differentiable part, allowing integration of gradient- or Newton-type methods.
4. Practical Implementations:
   • The algorithms can be employed in multidisciplinary applications ranging from genomics to radio astronomy.
   • Empirical studies show superior performance compared to established methods like FISTA and SpaRSA.

Numerical Results

The numerical experiments reinforce the paper’s claims by demonstrating the algorithm's superior performance on large-scale instances of the LASSO and logistic regression problems. These tests highlight the effectiveness of the algorithm's flexibility in adjusting the parallelism level and selectively updating sub-blocks, which are key to managing computational resources efficiently across different applications.

Future Directions

The paper sets a robust foundation for future research into distributed and parallel big data optimization. The methods outlined could be extended to more complex models involving additional constraints or more intricate data structures. Moreover, exploring adaptive strategies for tuning hyperparameters such as step-sizes and proximal-term coefficients could further improve the robustness and efficiency of the proposed framework in practice.

Implications

The theoretical and practical contributions of this research have significant implications for the field of optimization. By fusing parallel and selective update schemes with enhanced convergence guarantees, the framework presents a viable approach to tackle the challenges posed by nonconvex big data optimization problems. It opens avenues for more flexible algorithmic strategies crucial for scaling optimization tasks in various scientific and engineering domains.

Overall, the paper delivers a powerful toolset for both the academic community and industry practitioners, underscoring the potential of advanced nonlinear optimization techniques in addressing contemporary big data challenges.