Abstract: We study online reinforcement learning in linear Markov decision processes with adversarial losses and bandit feedback. We introduce two algorithms that achieve improved regret performance compared to existing approaches. The first algorithm, although computationally inefficient, achieves a regret of $\widetilde{O}(\sqrt{K})$ without relying on simulators, where $K$ is the number of episodes. This is the first rate-optimal result in the considered setting. The second algorithm is computationally efficient and achieves a regret of $\widetilde{O}(K^{\frac{3}{4}})$ . These results significantly improve over the prior state-of-the-art: a computationally inefficient algorithm by Kong et al. (2023) with $\widetilde{O}(K^{\frac{4}{5}}+1/\lambda_{\min})$ regret, and a computationally efficient algorithm by Sherman et al. (2023b) with $\widetilde{O}(K^{\frac{6}{7}})$ regret.