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In reinforcement learning, the reward function on current state and action iswidely used. When the objective is about the expectation of the (discounted)total reward only, it works perfectly. However, if the objective involves thetotal reward distribution, the result will be wrong. This paper studiesValue-at-Risk (VaR) problems in short- and long-horizon Markov decisionprocesses (MDPs) with two reward functions, which share the same expectations.Firstly we show that with VaR objective, when the real reward function istransition-based (with respect to action and both current and next states), thesimplified (state-based, with respect to action and current state only) rewardfunction will change the VaR. Secondly, for long-horizon MDPs, we estimate theVaR function with the aid of spectral theory and the central limit theorem.Thirdly, since the estimation method is for a Markov reward process with thereward function on current state only, we present a transformation algorithmfor the Markov reward process with the reward function on current and nextstates, in order to estimate the VaR function with an intact total rewarddistribution.

Transition-based versus State-based Reward Functions for MDPs with Value-at-Risk