  We present a polynomial time algorithm for online maximization of
$k$-submodular maximization. For online (nonmonotone) $k$-submodular
maximization, our algorithm achieves a tight approximate factor in an
approximate regret. For online monotone $k$-submodular maximization, our
approximate-regret matches to the best-known approximation ratio, which is
tight asymptotically as $k$ tends to infinity. Our approach is based on the
Blackwell approachability theorem and online linear optimization.
