  In this paper we study the adversarial combinatorial bandit with a known
non-linear reward function, extending existing work on adversarial linear
combinatorial bandit. {The adversarial combinatorial bandit with general
non-linear reward is an important open problem in bandit literature, and it is
still unclear whether there is a significant gap from the case of linear
reward, stochastic bandit, or semi-bandit feedback.} We show that, with $N$
arms and subsets of $K$ arms being chosen at each of $T$ time periods, the
minimax optimal regret is $\widetilde\Theta_{d}(\sqrt{N^d T})$ if the reward
function is a $d$-degree polynomial with $d< K$, and $\Theta_K(\sqrt{N^K T})$
if the reward function is not a low-degree polynomial. {Both bounds are
significantly different from the bound $O(\sqrt{\mathrm{poly}(N,K)T})$ for the
linear case, which suggests that there is a fundamental gap between the linear
and non-linear reward structures.} Our result also finds applications to
adversarial assortment optimization problem in online recommendation. We show
that in the worst-case of adversarial assortment problem, the optimal algorithm
must treat each individual $\binom{N}{K}$ assortment as independent.
