  We study the regret guarantee for risk-sensitive reinforcement learning
(RSRL) via distributional reinforcement learning (DRL) methods. In particular,
we consider finite episodic Markov decision processes whose objective is the
entropic risk measure (EntRM) of return. By leveraging a key property of the
EntRM, the independence property, we establish the risk-sensitive
distributional dynamic programming framework. We then propose two novel DRL
algorithms that implement optimism through two different schemes, including a
model-free one and a model-based one.
  We prove that they both attain $\tilde{\mathcal{O}}(\frac{\exp(|\beta|
H)-1}{|\beta|}H\sqrt{S^2AK})$ regret upper bound, where $S$, $A$, $K$, and $H$
represent the number of states, actions, episodes, and the time horizon,
respectively. It matches RSVI2 proposed in \cite{fei2021exponential}, with
novel distributional analysis. To the best of our knowledge, this is the first
regret analysis that bridges DRL and RSRL in terms of sample complexity.
  Acknowledging the computational inefficiency associated with the model-free
DRL algorithm, we propose an alternative DRL algorithm with distribution
representation. This approach not only maintains the established regret bounds
but also significantly amplifies computational efficiency.
  We also prove a tighter minimax lower bound of $\Omega(\frac{\exp(\beta
H/6)-1}{\beta H}H\sqrt{SAT})$ for the $\beta>0$ case, which recovers the tight
lower bound $\Omega(H\sqrt{SAT})$ in the risk-neutral setting.
