Asymptotically Optimal Regret for Reinforcement Learning without Horizon Dependence

We study horizon-free regret minimization for finite-horizon time-homogeneous tabular Markov decision processes with S states, A actions, horizon H, and per-trajectory total reward bounded by 1. We propose a new algorithm and prove a regret upper bound [tilde O(√SAK+S8A3)] with failure probability δ, where K is the number of episodes and tilde O(·) hides mathsfpolylog(S,A,K,1/δ). Thus, the regret is H-free and asymptotically optimal, matching the contextual-bandit lower bound Omega(√SAK) up to logarithmic factors. This completely removes the log H dependence from the previous tilde O(√SAKlog H+S2Alog H) guarantee of Zhang et al. (2021), and drastically improves the prior best horizon-free regret tilde O(√S9A3K) of Zhang et al. (2022) asymptotically. The main technical difficulty is that the optimal value functions Vh^*h=1H are time-inhomogeneous even though the transition kernel is time-homogeneous. A direct union bound over all value functions typically incurs an additional minlog H,S factor. We avoid this factor by (i) exploiting the monotonicity of Vh^* in h and (ii) non-trivially projecting the value functions onto an S-dimensional grid. Our analysis relies on three additional ingredients. First, we introduce a horizon-truncation argument that enables reward-based exploration and removes the cost of a separate reward-free exploration phase. Second, we design a cutting bonus that preserves both optimism and the monotonicity needed for planning. Third, we prove a new bound on total deviation for time-homogeneous MDPs, which controls the clipped variance terms in the cutting bonus with adjustable polynomial dependence on S and without any dependence on H. Together, these tools yield an asymptotically optimal horizon-free regret guarantee.