  In this paper, we consider the problem of prediction with expert advice in
dynamic environments. We choose tracking regret as the performance metric and
develop two adaptive and efficient algorithms with data-dependent tracking
regret bounds. The first algorithm achieves a second-order tracking regret
bound, which improves existing first-order bounds. The second algorithm enjoys
a path-length bound, which is generally not comparable to the second-order
bound but offers advantages in slowly moving environments. Both algorithms are
developed under the online mirror descent framework and draw inspiration from
existing algorithms that attain data-dependent bounds of static regret. The key
idea is to use a clipped simplex in the updating step of online mirror descent.
Finally, we extend our algorithms and analysis to online matrix prediction and
provide the first data-dependent tracking regret bound for this problem.
