  We consider dynamic multi-product pricing and assortment problems under an
unknown demand over T periods, where in each period, the seller decides on the
price for each product or the assortment of products to offer to a customer who
chooses according to an unknown Multinomial Logit Model (MNL). Such problems
arise in many applications, including online retail and advertising. We propose
a randomized dynamic pricing policy based on a variant of the Online Newton
Step algorithm (ONS) that achieves a $O(d\sqrt{T}\log(T))$ regret guarantee
under an adversarial arrival model. We also present a new optimistic algorithm
for the adversarial MNL contextual bandits problem, which achieves a better
dependency than the state-of-the-art algorithms in a problem-dependent constant
$\kappa_2$ (potentially exponentially small). Our regret upper bound scales as
$\tilde{O}(d\sqrt{\kappa_2 T}+ \log(T)/\kappa_2)$, which gives a stronger bound
than the existing $\tilde{O}(d\sqrt{T}/\kappa_2)$ guarantees.
