  We consider unreliable distributed learning systems wherein the training data
is kept confidential by external workers, and the learner has to interact
closely with those workers to train a model. In particular, we assume that
there exists a system adversary that can adaptively compromise some workers;
the compromised workers deviate from their local designed specifications by
sending out arbitrarily malicious messages.
  We assume in each communication round, up to $q$ out of the $m$ workers
suffer Byzantine faults. Each worker keeps a local sample of size $n$ and the
total sample size is $N=nm$. We propose a secured variant of the gradient
descent method that can tolerate up to a constant fraction of Byzantine
workers, i.e., $q/m = O(1)$. Moreover, we show the statistical estimation error
of the iterates converges in $O(\log N)$ rounds to $O(\sqrt{q/N} +
\sqrt{d/N})$, where $d$ is the model dimension. As long as $q=O(d)$, our
proposed algorithm achieves the optimal error rate $O(\sqrt{d/N})$. Our results
are obtained under some technical assumptions. Specifically, we assume
strongly-convex population risk. Nevertheless, the empirical risk (sample
version) is allowed to be non-convex. The core of our method is to robustly
aggregate the gradients computed by the workers based on the filtering
procedure proposed by Steinhardt et al. On the technical front, deviating from
the existing literature on robustly estimating a finite-dimensional mean
vector, we establish a {\em uniform} concentration of the sample covariance
matrix of gradients, and show that the aggregated gradient, as a function of
model parameter, converges uniformly to the true gradient function. To get a
near-optimal uniform concentration bound, we develop a new matrix concentration
inequality, which might be of independent interest.
