  We consider gradient coding in the presence of an adversary controlling
so-called malicious workers trying to corrupt the computations. Previous works
propose the use of MDS codes to treat the responses from malicious workers as
errors and correct them using the error-correction properties of the code. This
comes at the expense of increasing the replication, i.e., the number of workers
each partial gradient is computed by. In this work, we propose a way to reduce
the replication to $s+1$ instead of $2s+1$ in the presence of $s$ malicious
workers. Our method detects erroneous inputs from the malicious workers,
transforming them into erasures. This comes at the expense of $s$ additional
local computations at the main node and additional rounds of light
communication between the main node and the workers. We define a general
framework and give fundamental limits for fractional repetition data
allocations. Our scheme is optimal in terms of replication and local
computation and incurs a communication cost that is asymptotically, in the size
of the dataset, a multiplicative factor away from the derived bound. We
furthermore show how additional redundancy can be exploited to reduce the
number of local computations and communication cost, or, alternatively,
tolerate straggling workers.
