Large-scale distributed learning aims at minimizing a loss function $L$ that depends on a training dataset with respect to a $d$-length parameter vector. The distributed cluster typically consists of a parameter server (PS) and multiple workers. Gradient coding is a technique that makes the learning process resilient to straggling workers. It introduces redundancy within the assignment of data points to the workers and uses coding theoretic ideas so that the PS can recover $\nabla L$ exactly or approximately, even in the presence of stragglers. Communication-efficient gradient coding allows the workers to communicate vectors of length smaller than $d$ to the PS, thus reducing the communication time. While there have been schemes that address the exact recovery of $\nabla L$ within communication-efficient gradient coding, to the best of our knowledge the approximate variant has not been considered in a systematic manner. In this work we present constructions of communication-efficient approximate gradient coding schemes. Our schemes use structured matrices that arise from bipartite graphs, combinatorial designs and strongly regular graphs, along with randomization and algebraic constraints. We derive analytical upper bounds on the approximation error of our schemes that are tight in certain cases. Moreover, we derive a corresponding worst-case lower bound on the approximation error of any scheme. For a large class of our methods, under reasonable probabilistic worker failure models, we show that the expected value of the computed gradient equals the true gradient. This in turn allows us to prove that the learning algorithm converges to a stationary point over the iterations. Numerical experiments corroborate our theoretical findings.

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