  This paper studies safe Reinforcement Learning (safe RL) with linear function
approximation and under hard instantaneous constraints where unsafe actions
must be avoided at each step. Existing studies have considered safe RL with
hard instantaneous constraints, but their approaches rely on several key
assumptions: $(i)$ the RL agent knows a safe action set for {\it every} state
or knows a {\it safe graph} in which all the state-action-state triples are
safe, and $(ii)$ the constraint/cost functions are {\it linear}. In this paper,
we consider safe RL with instantaneous hard constraints without assumption
$(i)$ and generalize $(ii)$ to Reproducing Kernel Hilbert Space (RKHS). Our
proposed algorithm, LSVI-AE, achieves $\tilde{\cO}(\sqrt{d^3H^4K})$ regret and
$\tilde{\cO}(H \sqrt{dK})$ hard constraint violation when the cost function is
linear and $\cO(H\gamma_K \sqrt{K})$ hard constraint violation when the cost
function belongs to RKHS. Here $K$ is the learning horizon, $H$ is the length
of each episode, and $\gamma_K$ is the information gain w.r.t the kernel used
to approximate cost functions. Our results achieve the optimal dependency on
the learning horizon $K$, matching the lower bound we provide in this paper and
demonstrating the efficiency of LSVI-AE. Notably, the design of our approach
encourages aggressive policy exploration, providing a unique perspective on
safe RL with general cost functions and no prior knowledge of safe actions,
which may be of independent interest.
