Adaptive Private Distributed Matrix Multiplication

TL;DR

This paper introduces RPM3, a rateless private matrix multiplication scheme that adapts to worker heterogeneity, reducing waiting time at the expense of a lower rate compared to fixed-rate schemes.

Contribution

The paper presents RPM3, a novel rateless private matrix multiplication scheme that improves flexibility and reduces waiting time by leveraging worker heterogeneity.

Findings

RPM3 achieves smaller mean waiting time than fixed-rate schemes.

The scheme is adaptive and works efficiently in heterogeneous environments.

Upper and lower bounds for waiting time are provided under different worker models.

Abstract

We consider the problem of designing codes with flexible rate (referred to as rateless codes), for private distributed matrix-matrix multiplication. A master server owns two private matrices $\mathbf{A}$ and $\mathbf{B}$ and hires worker nodes to help computing their multiplication. The matrices should remain information-theoretically private from the workers. Codes with fixed rate require the master to assign tasks to the workers and then wait for a predetermined number of workers to finish their assigned tasks. The size of the tasks, hence the rate of the scheme, depends on the number of workers that the master waits for. We design a rateless private matrix-matrix multiplication scheme, called RPM3. In contrast to fixed-rate schemes, our scheme fixes the size of the tasks and allows the master to send multiple tasks to the workers. The master keeps sending tasks and receiving results until it can decode the multiplication; rendering the scheme flexible and adaptive to heterogeneous environments. Despite resulting in a smaller rate than known straggler-tolerant schemes, RPM3 provides a smaller mean waiting time of the master by leveraging the heterogeneity of the workers. The waiting time is studied under two different models for the workers' service time. We provide upper bounds for the mean waiting time under both models. In addition, we provide lower bounds on the mean waiting time under the worker-dependent fixed service time model.