Markov Decision Processes

A pivotal component of reinforcement learning involves understanding how decisions are made over time in uncertain environments. This chapter introduces Markov Decision Processes (MDPs), which serve as the mathematical framework for modeling decision-making problems. By capturing the dynamics of an environment through states, actions, and rewards, MDPs offer a structured approach to analyze and optimize the behavior of agents.

You will gain insights into the components of MDPs, including states, actions, transition probabilities, and reward functions. These elements form the basis for defining policies and value functions, key concepts that guide the decision-making process in uncertain scenarios. Furthermore, the chapter explores the notion of optimality, helping you understand how agents can strive for the best long-term outcomes.

Throughout the chapter, mathematical representations will be used to illustrate these ideas. For instance, the relationship between states and actions is captured through transition probabilities, denoted as $P(s'|s, a)$, which describe the likelihood of moving from state $s$ to state $s'$ after taking action $a$. Additionally, you will encounter the Bellman equation, a fundamental equation in reinforcement learning, presented as $V(s) = \sum_{a} \pi(a|s) \sum_{s'} P(s'|s, a) [R(s, a, s') + \gamma V(s')]$ which helps in understanding how future rewards are considered in decision-making.

By the end of this chapter, you will have a solid foundation in how MDPs underpin the algorithms and strategies used in reinforcement learning to achieve optimal decision-making.