  We study the sequential general online regression, known also as the
sequential probability assignments, under logarithmic loss when compared
against a broad class of experts. We focus on obtaining tight, often matching,
lower and upper bounds for the sequential minimax regret that are defined as
the excess loss it incurs over a class of experts. After proving a general
upper bound, we consider some specific classes of experts from Lipschitz class
to bounded Hessian class and derive matching lower and upper bounds with
provably optimal constants. Our bounds work for a wide range of values of the
data dimension and the number of rounds. To derive lower bounds, we use tools
from information theory (e.g., Shtarkov sum) and for upper bounds, we resort to
new "smooth truncated covering" of the class of experts. This allows us to find
constructive proofs by applying a simple and novel truncated Bayesian
algorithm. Our proofs are substantially simpler than the existing ones and yet
provide tighter (and often optimal) bounds.
