  We consider online convex optimization with time-varying stage costs and
additional switching costs. Since the switching costs introduce coupling across
all stages, multi-step-ahead (long-term) predictions are incorporated to
improve the online performance. However, longer-term predictions tend to suffer
from lower quality. Thus, a critical question is: how to reduce the impact of
long-term prediction errors on the online performance? To address this
question, we introduce a gradient-based online algorithm, Receding Horizon
Inexact Gradient (RHIG), and analyze its performance by dynamic regrets in
terms of the temporal variation of the environment and the prediction errors.
RHIG only considers at most $W$-step-ahead predictions to avoid being misled by
worse predictions in the longer term. The optimal choice of $W$ suggested by
our regret bounds depends on the tradeoff between the variation of the
environment and the prediction accuracy. Additionally, we apply RHIG to a
well-established stochastic prediction error model and provide expected regret
and concentration bounds under correlated prediction errors. Lastly, we
numerically test the performance of RHIG on quadrotor tracking problems.

Machine Learning;Systems and Control;Systems and Control;Optimization and Control