MDP Fundamentals

Markov Decision Processes (MDPs) play a pivotal role in reinforcement learning, encapsulating the essence of decision-making in uncertain environments. With a basic understanding of reinforcement learning concepts, we can delve deeper into the intricacies of MDPs, a powerful mathematical framework that models sequential decision-making.

At its core, an MDP is defined by a tuple $(S, A, P, R, \gamma)$, where each component serves a specific purpose in modeling the environment and the agent's interaction with it:

1. States (S): These represent all possible situations or configurations the environment can be in. For example, in a board game, a state might describe the entire configuration of the board at a given time.

2. Actions (A): These are the set of all possible moves or decisions the agent can make. Continuing with our board game analogy, an action could be moving a piece from one square to another.

3. Transition Probabilities (P): Denoted as $P(s'|s, a)$, these probabilities define the likelihood of moving from state $s$ to state $s'$ after taking action $a$. This captures the uncertainty inherent in many environments where the same action may lead to different outcomes.

4. Reward Function (R): The reward function assigns a numerical value to each state-action pair, often represented as $R(s, a, s')$. This value reflects the immediate benefit received after transitioning from state $s$ to state $s'$ due to action $a$. The reward function guides the agent towards desirable outcomes.

5. Discount Factor ($\gamma$): A crucial parameter that determines the importance of future rewards. It ranges between 0 and 1, where a value closer to 0 makes the agent short-sighted by preferring immediate rewards, while a value near 1 encourages the agent to consider long-term gains.

Diagram showing the components of a Markov Decision Process (MDP)

With these components, MDPs provide a structured approach to evaluate and optimize the decision-making strategy of an agent. The goal is to find an optimal policy $\pi$, which is a mapping from states to actions that maximizes the expected cumulative reward over time. This is where the concept of value functions becomes indispensable.

Value Functions: Two primary value functions are used in MDPs: the state-value function $V(s)$ and the action-value function $Q(s, a)$. The state-value function estimates the expected return (cumulative reward) starting from state $s$ and following a policy $\pi$, while the action-value function estimates the expected return starting from state $s$, taking action $a$, and thereafter following policy $\pi$.

One of the cornerstones of reinforcement learning is the Bellman Equation, which provides a recursive decomposition of the value function. For a given policy $\pi$, the Bellman equation for the state-value function is expressed as:

$V(s) = \sum_{a} \pi(a|s) \sum_{s'} P(s'|s, a) \left[ R(s, a, s') + \gamma V(s') \right]$

This equation states that the value of a state $s$ under policy $\pi$ equals the expected sum of the immediate reward and the discounted value of the next state, averaged over all possible actions and resulting states.

Optimality and Policies: The ultimate objective in MDPs is to find an optimal policy $\pi^*$ that yields the maximum value for each state. This leads us to the concept of the Optimal Value Function $V^*(s)$, which satisfies the Bellman Optimality Equation:

$V^*(s) = \max_{a} \sum_{s'} P(s'|s, a) \left[ R(s, a, s') + \gamma V^*(s') \right]$

By solving this equation, either analytically or through iterative methods, we can derive policies that ensure the agent acts in the most beneficial way possible in any given state.

Line chart comparing the state-value function V(s) and the optimal value function V*(s) across different states

By understanding these MDP fundamentals, you gain insight into how reinforcement learning algorithms are designed to find optimal strategies in complex environments. This knowledge sets the stage for exploring more advanced topics such as policy iteration, value iteration, and various approximation techniques that handle real-world applications where the state and action spaces are large or continuous.