# Improved Strong Worst-case Upper Bounds for MDP Planning

## Anchit Gupta, Shivaram Kalyanakrishnan

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence
Main track. Pages 1788-1794. https://doi.org/10.24963/ijcai.2017/248

The Markov Decision Problem (MDP) plays a central role in AI as an abstraction of sequential decision making. We contribute to the theoretical analysis of MDP PLANNING, which is the problem of computing an optimal policy for a given MDP. Specifically, we furnish improved STRONG WORST-CASE upper bounds on the running time of MDP planning. Strong bounds are those that depend only on the number of states n and the number of actions k in the specified MDP; they have no dependence on affiliated variables such as the discount factor and the number of bits needed to represent the MDP. Worst-case bounds apply to EVERY run of an algorithm; randomised algorithms can typically yield faster EXPECTED running times. While the special case of 2-action MDPs (that is, k = 2) has recently received some attention, bounds for general k have remained to be improved for several decades. Our contributions are to this general case. For k >= 3, the tightest strong upper bound shown to date for MDP planning belongs to a family of algorithms called Policy Iteration. This bound is only a polynomial improvement over a trivial bound of poly(n, k) k^{n} [Mansour and Singh, 1999]. In this paper, we generalise a contrasting algorithm called the Fibonacci Seesaw, and derive a bound of poly(n, k) k^{0.6834n}. The key construct we use is a template to map algorithms for the 2-action setting to the general setting. Interestingly, this idea can also be used to design Policy Iteration algorithms with a running time upper bound of poly(n, k) k^{0.7207n}. Both our results improve upon bounds that have stood for several decades.
Keywords:
Machine Learning: Reinforcement Learning
Planning and Scheduling: Markov Decisions Processes
Planning and Scheduling: Planning Algorithms
Uncertainty in AI: Sequential Decision Making