We consider the learning problem under an online Markov decision process (MDP) aimed at learning the time-dependent decision-making policy of an agent that minimizes the regret—the difference from the best fixed policy. The difficulty of online MDP learning is that the reward function changes over time. In this letter, we show that a simple online policy gradient algorithm achieves regret for T steps under a certain concavity assumption and under a strong concavity assumption. To the best of our knowledge, this is the first work to present an online MDP algorithm that can handle continuous state, action, and parameter spaces with guarantee. We also illustrate the behavior of the proposed online policy gradient method through experiments.

The policy gradient approach is a flexible and powerful reinforcement learning method particularly for problems with continuous actions such as robot control. A common challenge is how to reduce the variance of policy gradient estimates for reliable policy updates. In this letter, we combine the following three ideas and give a highly effective policy gradient method: (1) policy gradients with parameter-based exploration, a recently proposed policy search method with low variance of gradient estimates; (2) an importance sampling technique, which allows us to reuse previously gathered data in a consistent way; and (3) an optimal baseline, which minimizes the variance of gradient estimates with their unbiasedness being maintained. For the proposed method, we give a theoretical analysis of the variance of gradient estimates and show its usefulness through extensive experiments.