Memoryless Exact Solutions for Deterministic MDPs with Sparse Rewards
This work addresses the challenge of scalability in MDPs for domains with sparse rewards, offering an exact solution that avoids state space explosion, which is incremental as it builds on existing MDP methods but introduces efficiency improvements.
The authors tackled the problem of solving deterministic continuous Markov Decision Processes with sparse rewards by proposing an algorithm that computes the optimal policy exactly, independent of state space size, with time complexity O(|R|^3 × |A|^2) and memory complexity O(|R| × |A|). They also introduced a companion algorithm for following the optimal policy on-demand, linear in path length, and demonstrated it alongside value iteration on tractable MDPs.
We propose an algorithm for deterministic continuous Markov Decision Processes with sparse rewards that computes the optimal policy exactly with no dependency on the size of the state space. The algorithm has time complexity of $O( |R|^3 \times |A|^2 )$ and memory complexity of $O( |R| \times |A| )$, where $|R|$ is the number of reward sources and $|A|$ is the number of actions. Furthermore, we describe a companion algorithm that can follow the optimal policy from any initial state without computing the entire value function, instead computing on-demand the value of states as they are needed. The algorithm to solve the MDP does not depend on the size of the state space for either time or memory complexity, and the ability to follow the optimal policy is linear in time and space with the path length of following the optimal policy from the initial state. We demonstrate the algorithm operation side by side with value iteration on tractable MDPs.