  A natural goal when designing online learning algorithms for non-stationary
environments is to bound the regret of the algorithm in terms of the temporal
variation of the input sequence. Intuitively, when the variation is small, it
should be easier for the algorithm to achieve low regret, since past
observations are predictive of future inputs. Such data-dependent "pathlength"
regret bounds have recently been obtained for a wide variety of online learning
problems, including OCO and bandits. We obtain the first pathlength regret
bounds for online control and estimation (e.g. Kalman filtering) in linear
dynamical systems. The key idea in our derivation is to reduce
pathlength-optimal filtering and control to certain variational problems in
robust estimation and control; these reductions may be of independent interest.
Numerical simulations confirm that our pathlength-optimal algorithms outperform
traditional $H_2$ and $H_{\infty}$ algorithms when the environment varies over
time.
