Polynomial Codes: an Optimal Design for High-Dimensional Coded Matrix Multiplication (1705.10464v2)

Abstract: We consider a large-scale matrix multiplication problem where the computation is carried out using a distributed system with a master node and multiple worker nodes, where each worker can store parts of the input matrices. We propose a computation strategy that leverages ideas from coding theory to design intermediate computations at the worker nodes, in order to efficiently deal with straggling workers. The proposed strategy, named as \emph{polynomial codes}, achieves the optimum recovery threshold, defined as the minimum number of workers that the master needs to wait for in order to compute the output. Furthermore, by leveraging the algebraic structure of polynomial codes, we can map the reconstruction problem of the final output to a polynomial interpolation problem, which can be solved efficiently. Polynomial codes provide order-wise improvement over the state of the art in terms of recovery threshold, and are also optimal in terms of several other metrics. Furthermore, we extend this code to distributed convolution and show its order-wise optimality.

• The paper introduces polynomial codes to optimize distributed matrix multiplication by achieving a minimum recovery threshold that mitigates straggler effects.
• It leverages polynomial interpolation to balance workload across nodes and reduce redundant computations in large-scale distributed systems.
• Empirical results show decreased latency and improved fault tolerance, with extensions to distributed convolution indicating broad practical applications.

Evaluation of Polynomial Codes for High-Dimensional Coded Matrix Multiplication

The paper presents an advanced investigation into distributed matrix multiplication by integrating concepts from coding theory. The authors address a critical performance bottleneck in distributed computing: the latency caused by "straggler" nodes, which are slower in completing tasks. By introducing the concept of polynomial codes, the researchers propose an innovative approach to overseeing the computation across distributed systems, achieving optimal recovery thresholds to mitigate the impact of these stragglers.

The principal contribution of the paper is the introduction of polynomial codes which leverage the algebraic structure of input data to minimize the number of worker nodes that are required to complete their tasks to reconstruct the final matrix multiplication product. This approach specifically addresses the issue of redundant computations by optimally balancing the workload across available nodes. Consequently, the polynomial codes achieve an impressive theoretical minimum recovery threshold of $mn$, independent of the number of nodes $N$, and improve upon previous state-of-the-art coding methods that presented recovery thresholds scaling with $N$ or $\sqrt{N}$.

The authors provide an array of empirical results, demonstrating the efficacy of polynomial codes in decreasing computation latency and enhancing fault tolerance. By constructing a mapping between the distributed matrix multiplication problem and polynomial interpolation, a computationally efficient decoding algorithm is achieved. This correlation to polynomial interpolation establishes a foundation for reconstruction that is computationally feasible, especially beneficial for large datasets typical in modern data-intensive applications.

Significantly, the paper extends polynomial codes to the domain of distributed convolution, a related problem with wide applications across engineering and scientific computing tasks. The polynomial codes are adapted to achieve a further reduced recovery threshold, specifically $m + n - 1$, and the authors underline that their strategy holds potential for optimal performance within a factor of $2$ concerning distributed convolution, offering an intriguing avenue for further research into finding exact thresholds.

From a theoretical perspective, the introduction of polynomial coding techniques for distributed matrix multiplication tackles foundational issues in distributed computing networks, especially under stringent resource constraints. Practically, the paper indicates significant implications for systems relying on large-scale matrix operations, such as those utilized in machine learning algorithms and large-scale simulations.

Future work could focus on exploring enhanced polynomial interpolation techniques that might yield more efficient implementations or provide tighter optimality bounds on recovery thresholds across varying problem classes. Additionally, investigating different forms of redundancy or leveraging alternative algebraic structures could extend polynomial codes' applicability and performance in diverse distributed environments.

In summary, this research contributes substantively to the field of distributed computing by not only offering theoretical advancements in the matrix multiplication domain but also by suggesting practical frameworks that can be harnessed to manage computational resources more efficiently, with significant implications for the scalability and reliability of distributed processing systems.