  We consider the problem of Byzantine fault-tolerance in the peer-to-peer
(P2P) distributed gradient-descent method -- a prominent algorithm for
distributed optimization in a P2P system. In this problem, the system comprises
of multiple agents, and each agent has a local cost function. In the fault-free
case, when all the agents are honest, the P2P distributed gradient-descent
method allows all the agents to reach a consensus on a solution that minimizes
their aggregate cost. However, we consider a scenario where a certain number of
agents may be Byzantine faulty. Such faulty agents may not follow an algorithm
correctly, and may share arbitrary incorrect information to prevent other
non-faulty agents from solving the optimization problem. In the presence of
Byzantine faulty agents, a more reasonable goal is to allow all the non-faulty
agents to reach a consensus on a solution that minimizes the aggregate cost of
all the non-faulty agents. We refer to this fault-tolerance goal as
$f$-resilience where $f$ is the maximum number of Byzantine faulty agents in a
system of $n$ agents, with $f < n$. Most prior work on fault-tolerance in P2P
distributed optimization only consider approximate fault-tolerance wherein,
unlike $f$-resilience, all the non-faulty agents' compute a minimum point of a
non-uniformly weighted aggregate of their cost functions. We propose a
fault-tolerance mechanism that confers provable $f$-resilience to the P2P
distributed gradient-descent method, provided the non-faulty agents satisfy the
necessary condition of $2f$-redundancy, defined later in the paper. Moreover,
compared to prior work, our algorithm is applicable to a larger class of
high-dimensional convex distributed optimization problems.
