# On Bellman's principle of optimality and Reinforcement learning for safety-constrained Markov decision process

## TL;DR

This paper investigates optimality principles in safety-constrained Markov decision processes, revealing limitations of Bellman's principle and proposing a game-theoretic approach with a new Q-learning algorithm for safe reinforcement learning.

## Contribution

It demonstrates that Bellman's principle may not hold in multichain constrained MDPs and introduces a zero-sum game formulation with a novel asynchronous value iteration and Q-learning method.

## Key findings

- Bellman's principle does not always apply to constrained MDPs with multichain structures
- A zero-sum game formulation enables solving the multi-objective optimization problem
- A modified Q-learning algorithm with error bounds is proposed for safe reinforcement learning.

## Abstract

We study optimality for the safety-constrained Markov decision process which is the underlying framework for safe reinforcement learning. Specifically, we consider a constrained Markov decision process (with finite states and finite actions) where the goal of the decision maker is to reach a target set while avoiding an unsafe set(s) with certain probabilistic guarantees. Therefore the underlying Markov chain for any control policy will be multichain since by definition there exists a target set and an unsafe set. The decision maker also has to be optimal (with respect to a cost function) while navigating to the target set. This gives rise to a multi-objective optimization problem. We highlight the fact that Bellman's principle of optimality may not hold for constrained Markov decision problems with an underlying multichain structure (as shown by the counterexample due to Haviv. We resolve the counterexample by formulating the aforementioned multi-objective optimization problem as a zero-sum game and thereafter construct an asynchronous value iteration scheme for the Lagrangian (similar to Shapley's algorithm). Finally, we consider the reinforcement learning problem for the same and construct a modified $Q$-learning algorithm for learning the Lagrangian from data. We also provide a lower bound on the number of iterations required for learning the Lagrangian and corresponding error bounds.