Convergence of SARSA with linear function approximation: The random horizon case

The reinforcement learning algorithm SARSA combined with linear function approximation has been shown to converge for infinite horizon discounted Markov decision problems (MDPs). In this paper, we investigate the convergence of the algorithm for random horizon MDPs, which has not previously been shown. We show, similar to earlier results for infinite horizon discounted MDPs, that if the behaviour policy is $\varepsilon$-soft and Lipschitz continuous with respect to the weight vector of the linear function approximation, with small enough Lipschitz constant, then the algorithm will converge with probability one when considering a random horizon MDP.