  We investigate the problem of online learning, which has gained significant
attention in recent years due to its applicability in a wide range of fields
from machine learning to game theory. Specifically, we study the online
optimization of mixable loss functions with logarithmic static regret in a
dynamic environment. The best dynamic estimation sequence that we compete
against is selected in hindsight with full observation of the loss functions
and is allowed to select different optimal estimations in different time
intervals (segments). We propose an online mixture framework that uses these
static solvers as the base algorithm. We show that with the suitable selection
of hyper-expert creations and weighting strategies, we can achieve logarithmic
and squared logarithmic regret per switch in quadratic and linearithmic
computational complexity, respectively. For the first time in literature, we
show that it is also possible to achieve near-logarithmic regret per switch
with sub-polynomial complexity per time. Our results are guaranteed to hold in
a strong deterministic sense in an individual sequence manner.
