  Non-stationary online learning has drawn much attention in recent years. In
particular, dynamic regret and adaptive regret are proposed as two principled
performance measures for online convex optimization in non-stationary
environments. To optimize them, a two-layer online ensemble is usually deployed
due to the inherent uncertainty of the non-stationarity, in which a group of
base-learners are maintained and a meta-algorithm is employed to track the best
one on the fly. However, the two-layer structure raises the concern about the
computational complexity -- those methods typically maintain $\mathcal{O}(\log
T)$ base-learners simultaneously for a $T$-round online game and thus perform
multiple projections onto the feasible domain per round, which becomes the
computational bottleneck when the domain is complicated. In this paper, we
present efficient methods for optimizing dynamic regret and adaptive regret,
which reduce the number of projections per round from $\mathcal{O}(\log T)$ to
$1$. Moreover, our obtained algorithms require only one gradient query and one
function evaluation at each round. Our technique hinges on the reduction
mechanism developed in parameter-free online learning and requires non-trivial
twists on non-stationary online methods. Empirical studies verify our
theoretical findings.
