Probability density estimation of the Q Function for reinforcement learning

Performing Q-Learning in continuous state-action spaces is a problem still unsolved for many complex applications. The Q function may be rather complex and can not be expected to fit into a predefined parametric model. In addition, the function approximation must be able to cope with the high non-stationarity of the estimated q values, the on-line nature of the learning with a strongly biased sampling to convergence regions, and the large amount of generalization required for a feasible implementation. To cope with these problems local, non-parametric function approximations seem more suitable than global parametric ones. A kind of function approximation that is gaining special interest in the field of machine learning are those based on densities. Estimating densities provides more information than simple function approximations which can be used to deal with the Reinforcement Learning problems. For instance, density estimation permits to know the actual distribution of the q values for any given state-action, and provides information about how many data has been collected in different regions of the domain. In this work we propose a Q-Learning approach for continuous state-action spaces based on joint density estimations. The density distribution is represented with a Gaussian Mixture Model using an on-line version of the Expectation-Maximization algorithm. We propose a method that handles the biased sampling problem with good performance. Experiments performed on a test problem show remarkable improvements over previous published results