  Gradient coding schemes effectively mitigate full stragglers in distributed
learning by introducing identical redundancy in coded local partial derivatives
corresponding to all model parameters. However, they are no longer effective
for partial stragglers as they cannot utilize incomplete computation results
from partial stragglers. This paper aims to design a new gradient coding scheme
for mitigating partial stragglers in distributed learning. Specifically, we
consider a distributed system consisting of one master and N workers,
characterized by a general partial straggler model and focuses on solving a
general large-scale machine learning problem with L model parameters using
gradient coding. First, we propose a coordinate gradient coding scheme with L
coding parameters representing L possibly different diversities for the L
coordinates, which generates most gradient coding schemes. Then, we consider
the minimization of the expected overall runtime and the maximization of the
completion probability with respect to the L coding parameters for coordinates,
which are challenging discrete optimization problems. To reduce computational
complexity, we first transform each to an equivalent but much simpler discrete
problem with N\llL variables representing the partition of the L coordinates
into N blocks, each with identical redundancy. This indicates an equivalent but
more easily implemented block coordinate gradient coding scheme with N coding
parameters for blocks. Then, we adopt continuous relaxation to further reduce
computational complexity. For the resulting minimization of expected overall
runtime, we develop an iterative algorithm of computational complexity O(N^2)
to obtain an optimal solution and derive two closed-form approximate solutions
both with computational complexity O(N). For the resultant maximization of the
completion probability, we develop an iterative algorithm of...
