Parallel Coordinate Descent Methods for Big Data Optimization (1212.0873v2)

Abstract: In this work we show that randomized (block) coordinate descent methods can be accelerated by parallelization when applied to the problem of minimizing the sum of a partially separable smooth convex function and a simple separable convex function. The theoretical speedup, as compared to the serial method, and referring to the number of iterations needed to approximately solve the problem with high probability, is a simple expression depending on the number of parallel processors and a natural and easily computable measure of separability of the smooth component of the objective function. In the worst case, when no degree of separability is present, there may be no speedup; in the best case, when the problem is separable, the speedup is equal to the number of processors. Our analysis also works in the mode when the number of blocks being updated at each iteration is random, which allows for modeling situations with busy or unreliable processors. We show that our algorithm is able to solve a LASSO problem involving a matrix with 20 billion nonzeros in 2 hours on a large memory node with 24 cores.

• The paper introduces parallelized randomized block coordinate descent, highlighting its efficacy in speeding up composite convex optimization.
• It establishes a theoretical speedup formula linking the number of processors to the separability degree of the problem.
• A LASSO problem test with 20 billion nonzeros on 24 cores underscores the method’s scalability and practical efficiency.

Parallel Coordinate Descent Methods for Big Data Optimization

This paper, authored by Peter Richtárik and Martin Takáč, explores the parallel acceleration of randomized block coordinate descent methods within the context of optimizing the sum of a partially separable smooth convex function and a simple separable convex function. The primary focus is on demonstrating a speedup, achieved through parallelization, when comparing these methods to their serial counterparts.

Key Findings and Numerical Results

• Randomized Block Coordinate Descent Benefits: The authors establish that implementing parallelization in randomized block coordinate descent methods can lead to significant acceleration in solving optimization problems. This is particularly effective when minimizing a composite objective function, composed of a partially separable smooth convex function and a simple separable convex function.
• Theoretical Speedup: The paper provides a theoretical expression for speedup, dependent on the number of processors and the degree of separability within the smooth part of the objective function. Notably, if the problem is fully separable, the speedup is maximal, matching the number of processors used.
• Modeled Variability in Block Updates: The research accounts for variable block updates at each iteration, modeling scenarios with unreliable or busy processors. This is a critical adaptation for parallel computation scenarios where processor availability may vary unpredictably.
• Practical Demonstration: An experiment showcased the ability to solve a LASSO problem with a data matrix of 20 billion nonzeros in just two hours on a node equipped with 24 cores, illustrating the method's applicability to problems of substantial size and complexity.

Implications and Future Prospects

The paper firmly positions parallel coordinate descent methods as a highly feasible approach for large-scale optimization tasks, where resource efficiency and computational scalability are paramount. The findings have practical implications for a variety of domains, including machine learning, network analysis, and compressed sensing, where big data challenges necessitate robust, scalable optimization algorithms.

1. Scalability: The demonstrated efficiency in handling massive datasets makes these methods compelling for industries dealing with continually growing data sizes.
2. Flexibility and Robustness: The flexibility in accommodating uncertain processing power through stochastic block updates broadens the applicability of these methods, equipping them for dynamic computational environments.
3. Theoretical Insights: The theoretical advances in expected separable overapproximation (ESO) provide a foundational mathematical framework that could guide further research into optimizing parallel computations.
4. Future Research Directions: Further exploration is warranted into extending these methods for non-convex problems and integrating them with other forms of large-scale optimization techniques, possibly enhancing adaptive mechanisms for processor variability.

In conclusion, the paper's profound exploration of parallel coordinate descent methods holds promise for optimizing large-scale, complex systems, aligning well with contemporary computational demands in big data environments. The authors' contributions significantly advance both the theoretical understanding and practical implementation of these methodologies.