Variance Reduced Value Iteration and Faster Algorithms for Solving Markov Decision Processes
This work provides faster algorithms for solving MDPs, which is a core problem in reinforcement learning and decision-making, with incremental improvements over previous methods in specific parameter regimes.
The paper tackles the problem of solving discounted Markov Decision Processes (DMDPs) more efficiently by introducing variance reduction techniques, resulting in algorithms that compute ε-optimal policies with improved time complexities, such as nearly linear time for intermediate discount factors and sublinear time under sampling assumptions.
In this paper we provide faster algorithms for approximately solving discounted Markov Decision Processes in multiple parameter regimes. Given a discounted Markov Decision Process (DMDP) with $|S|$ states, $|A|$ actions, discount factor $γ\in(0,1)$, and rewards in the range $[-M, M]$, we show how to compute an $ε$-optimal policy, with probability $1 - δ$ in time \[ \tilde{O}\left( \left(|S|^2 |A| + \frac{|S| |A|}{(1 - γ)^3} \right) \log\left( \frac{M}ε \right) \log\left( \frac{1}δ \right) \right) ~ . \] This contribution reflects the first nearly linear time, nearly linearly convergent algorithm for solving DMDPs for intermediate values of $γ$. We also show how to obtain improved sublinear time algorithms provided we can sample from the transition function in $O(1)$ time. Under this assumption we provide an algorithm which computes an $ε$-optimal policy with probability $1 - δ$ in time \[ \tilde{O} \left(\frac{|S| |A| M^2}{(1 - γ)^4 ε^2} \log \left(\frac{1}δ\right) \right) ~. \] Lastly, we extend both these algorithms to solve finite horizon MDPs. Our algorithms improve upon the previous best for approximately computing optimal policies for fixed-horizon MDPs in multiple parameter regimes. Interestingly, we obtain our results by a careful modification of approximate value iteration. We show how to combine classic approximate value iteration analysis with new techniques in variance reduction. Our fastest algorithms leverage further insights to ensure that our algorithms make monotonic progress towards the optimal value. This paper is one of few instances in using sampling to obtain a linearly convergent linear programming algorithm and we hope that the analysis may be useful more broadly.