  We study online optimization in a setting where an online learner seeks to
optimize a per-round hitting cost, which may be non-convex, while incurring a
movement cost when changing actions between rounds. We ask: \textit{under what
general conditions is it possible for an online learner to leverage predictions
of future cost functions in order to achieve near-optimal costs?} Prior work
has provided near-optimal online algorithms for specific combinations of
assumptions about hitting and switching costs, but no general results are
known. In this work, we give two general sufficient conditions that specify a
relationship between the hitting and movement costs which guarantees that a new
algorithm, Synchronized Fixed Horizon Control (SFHC), provides a $1+O(1/w)$
competitive ratio, where $w$ is the number of predictions available to the
learner. Our conditions do not require the cost functions to be convex, and we
also derive competitive ratio results for non-convex hitting and movement
costs. Our results provide the first constant, dimension-free competitive ratio
for online non-convex optimization with movement costs. Further, we give an
example of a natural instance, Convex Body Chasing (CBC), where the sufficient
conditions are not satisfied and we can prove that no online algorithm can have
a competitive ratio that converges to 1.
