Value Function in Frequency Domain and the Characteristic Value Iteration Algorithm

This paper considers the problem of estimating the distribution of returns in reinforcement learning (i.e., distributional RL problem). It presents a new representational framework to maintain the uncertainty of returns and provides mathematical tools to compute it. We show that instead of representing a probability distribution function of returns, one can represent their characteristic function instead, the Fourier transform of their distribution. We call the new representation Characteristic Value Function (CVF), which can be interpreted as the frequency domain representation of the probability distribution of returns. We show that the CVF satisfies a Bellman-like equation, and its corresponding Bellman operator is contraction with respect to certain metrics. The contraction property allows us to devise an iterative procedure to compute the CVF, which we call Characteristic Value Iteration (CVI). We analyze CVI and its approximate variant and show how approximation errors affect the quality of computed CVF.