  Optimization in the presence of sharp (non-Lipschitz), unpredictable (w.r.t.
time and amount) changes is a challenging and largely unexplored problem of
great significance. We consider the class of piecewise Lipschitz functions,
which is the most general online setting considered in the literature for the
problem, and arises naturally in various combinatorial algorithm selection
problems where utility functions can have sharp discontinuities. The usual
performance metric of $\mathit{static}$ regret minimizes the gap between the
payoff accumulated and that of the best fixed point for the entire duration,
and thus fails to capture changing environments. Shifting regret is a useful
alternative, which allows for up to $s$ environment shifts. In this work we
provide an $O(\sqrt{sdT\log T}+sT^{1-\beta})$ regret bound for
$\beta$-dispersed functions, where $\beta$ roughly quantifies the rate at which
discontinuities appear in the utility functions in expectation (typically
$\beta\ge1/2$ in problems of practical interest). We also present a lower bound
tight up to sub-logarithmic factors. We further obtain improved bounds when
selecting from a small pool of experts. We empirically demonstrate a key
application of our algorithms to online clustering problems on popular
benchmarks.
