  We consider online algorithms under both the competitive ratio criteria and
the regret minimization one. Our main goal is to build a unified methodology
that would be able to guarantee both criteria simultaneously.
  For a general class of online algorithms, namely any Metrical Task System
(MTS), we show that one can simultaneously guarantee the best known competitive
ratio and a natural regret bound. For the paging problem we further show an
efficient online algorithm (polynomial in the number of pages) with this
guarantee.
  To this end, we extend an existing regret minimization algorithm
(specifically, Kapralov and Panigrahy) to handle movement cost (the cost of
switching between states of the online system). We then show how to use the
extended regret minimization algorithm to combine multiple online algorithms.
Our end result is an online algorithm that can combine a "base" online
algorithm, having a guaranteed competitive ratio, with a range of online
algorithms that guarantee a small regret over any interval of time. The
combined algorithm guarantees both that the competitive ratio matches that of
the base algorithm and a low regret over any time interval.
  As a by product, we obtain an expert algorithm with close to optimal regret
bound on every time interval, even in the presence of switching costs. This
result is of independent interest.
