  We study online reinforcement learning in linear Markov decision processes
with adversarial losses and bandit feedback, without prior knowledge on
transitions or access to simulators. We introduce two algorithms that achieve
improved regret performance compared to existing approaches. The first
algorithm, although computationally inefficient, ensures a regret of
$\widetilde{\mathcal{O}}\left(\sqrt{K}\right)$, where $K$ is the number of
episodes. This is the first result with the optimal $K$ dependence in the
considered setting. The second algorithm, which is based on the policy
optimization framework, guarantees a regret of
$\widetilde{\mathcal{O}}\left(K^{\frac{3}{4}} \right)$ and is computationally
efficient. Both our results significantly improve over the state-of-the-art: a
computationally inefficient algorithm by Kong et al. [2023] with
$\widetilde{\mathcal{O}}\left(K^{\frac{4}{5}}+poly\left(\frac{1}{\lambda_{\min}}\right)
\right)$ regret, for some problem-dependent constant $\lambda_{\min}$ that can
be arbitrarily close to zero, and a computationally efficient algorithm by
Sherman et al. [2023b] with $\widetilde{\mathcal{O}}\left(K^{\frac{6}{7}}
\right)$ regret.
