  We study online convex optimization under stochastic sub-gradient observation
faults, where we introduce adaptive algorithms with minimax optimal regret
guarantees. We specifically study scenarios where our sub-gradient observations
can be noisy or even completely missing in a stochastic manner. To this end, we
propose algorithms based on sub-gradient descent method, which achieve tight
minimax optimal regret bounds. When necessary, these algorithms utilize
properties of the underlying stochastic settings to optimize their learning
rates (step sizes). These optimizations are the main factor in providing the
minimax optimal performance guarantees, especially when observations are
stochastically missing. However, in real world scenarios, these properties of
the underlying stochastic settings may not be revealed to the optimizer. For
such a scenario, we propose a blind algorithm that estimates these properties
empirically in a generally applicable manner. Through extensive experiments, we
show that this empirical approach is a natural combination of regular
stochastic gradient descent and the minimax optimal algorithms (which work best
for randomized and adversarial function sequences, respectively).
