  We propose algorithms for online principal component analysis (PCA) and
variance minimization for adaptive settings. Previous literature has focused on
upper bounding the static adversarial regret, whose comparator is the optimal
fixed action in hindsight. However, static regret is not an appropriate metric
when the underlying environment is changing. Instead, we adopt the adaptive
regret metric from the previous literature and propose online adaptive
algorithms for PCA and variance minimization, that have sub-linear adaptive
regret guarantees. We demonstrate both theoretically and experimentally that
the proposed algorithms can adapt to the changing environments.
