  We study the problem of Online Convex Optimization (OCO) with memory, which
allows loss functions to depend on past decisions and thus captures temporal
effects of learning problems. In this paper, we introduce dynamic policy regret
as the performance measure to design algorithms robust to non-stationary
environments, which competes algorithms' decisions with a sequence of changing
comparators. We propose a novel algorithm for OCO with memory that provably
enjoys an optimal dynamic policy regret in terms of time horizon,
non-stationarity measure, and memory length. The key technical challenge is how
to control the switching cost, the cumulative movements of player's decisions,
which is neatly addressed by a novel switching-cost-aware online ensemble
approach equipped with a new meta-base decomposition of dynamic policy regret
and a careful design of meta-learner and base-learner that explicitly
regularizes the switching cost. The results are further applied to tackle
non-stationarity in online non-stochastic control (Agarwal et al., 2019), i.e.,
controlling a linear dynamical system with adversarial disturbance and convex
cost functions. We derive a novel gradient-based controller with dynamic policy
regret guarantees, which is the first controller provably competitive to a
sequence of changing policies for online non-stochastic control.
