Lecture 2: Markov Decision Process | Highlights and Annotations by Gistr.

MDPs model sequential decision-making using states, actions, and rewards. The Markov property (future depends only on the present) is key. Value functions represent expected future rewards, calculated via the Bellman equation. Optimal policies maximize these rewards, solvable iteratively (e.g., value/policy iteration). The lecture uses examples to illustrate these concepts. This segment delves into the Markov property, a fundamental concept in MDPs. It explains that the Markov property states that the future state of the system depends only on the current state and not on the past history. This property simplifies the modeling of complex systems by allowing us to focus only on the present state when predicting the future. The explanation clarifies how this property enables efficient representation and analysis of the system's dynamics. This segment introduces the concept of Markov Decision Processes (MDPs) as a framework for reinforcement learning problems. It explains how MDPs represent the interaction between an agent and its environment, focusing on the fully observable case where the agent has complete knowledge of the environment's state. The explanation highlights the importance of MDPs as a unifying framework applicable to various reinforcement learning scenarios. This segment explains the intuitive reasons behind using a discount factor in reinforcement learning, primarily focusing on the uncertainty in future rewards and the imperfect modeling of the environment. It also touches upon the mathematical convenience and the natural application in financial settings where the present value of money is higher than its future value. This segment uses a concrete example – a student's experience in a class – to illustrate the concept of a Markov process. It shows how to model the student's transitions between different states (e.g., attending class, checking Facebook, sleeping) using a transition matrix. The example demonstrates how to visualize and interpret the transition probabilities and generate sample sequences representing the student's behavior. The discussion also addresses the handling of non-stationary dynamics, where probabilities change over time.This segment extends the Markov process to include rewards and discount factors, creating a Markov reward process. It explains the reward function, which assigns a value to each state, and the discount factor, which determines the present value of future rewards. The explanation clarifies how these additions enable the formulation of reinforcement learning problems where the goal is to maximize the accumulated discounted reward over time. The concept of return (total accumulated reward) is introduced, and the role of the discount factor in making the return finite is discussed. This segment introduces the concept of the value function as the central quantity of interest in reinforcement learning. It defines the value function as the long-term value of being in a particular state, considering the expected return from that state onwards in a stochastic Markov reward process. The segment sets the stage for understanding how to evaluate the goodness of being in any given state.This segment demonstrates how to estimate the value function by sampling returns from the Markov reward process. It shows how to calculate the discounted return for a given sample sequence and explains how to estimate the value function by averaging over multiple sample returns. The distinction between the random nature of returns and the non-random nature of the value function (an expectation) is highlighted. This segment introduces the Bellman equation, a fundamental relationship in reinforcement learning that recursively decomposes the value function into immediate reward and the discounted value of the next state. The concept is explained intuitively using the analogy of a robot receiving immediate reward and then ending up in a new state, whose value needs to be considered. This segment utilizes backup diagrams to provide a visual understanding of the Bellman equation. It explains how the backup diagrams represent a one-step look-ahead search, averaging over possible outcomes to calculate the value function at a given state. This visualization aids in understanding the iterative nature of value function computation. This segment explains how to represent the Bellman equation using matrices and vectors, providing a concise formulation for calculating the value function of each state. This efficient representation is crucial for solving Markov Reward Processes (MRP), especially for larger state spaces, and serves as a foundation for understanding more complex methods. This segment introduces the concept of Markov Decision Processes (MDPs) by extending Markov Reward Processes (MRPs) to include actions and decisions. It explains how the addition of an action space allows for agency and decision-making, changing the problem from passively observing transitions to actively influencing them to maximize rewards.This segment defines policies as mappings from states to action probabilities and introduces two crucial value functions: the state value function (Vπ(s)) and the action value function (Qπ(s, a)). It clarifies how these functions help evaluate the long-term value of being in a state or taking an action under a given policy, which is essential for optimal decision-making in MDPs.This segment highlights the connection between Markov Decision Processes (MDPs) and Markov Reward Processes (MRPs), demonstrating that an MRP can be derived from an MDP given a fixed policy. This connection underscores the hierarchical relationship between the two models, showing how MDPs generalize MRPs by incorporating decision-making. This segment introduces the concept of optimal value functions (V* and Q*) in MDPs. It explains that V* represents the maximum possible reward achievable under any policy, while Q* shows the maximum reward achievable from a given state after taking a specific action. The explanation emphasizes the importance of Q* in determining the optimal policy because it directly indicates the best action to take in any given situation. This segment explains the core concept of the Bellman equation, a recursive relationship that defines the value of being in a particular state within an MDP. It uses a two-step look-ahead approach to illustrate how the value function relates to itself at the next step, averaging over actions and transition probabilities to determine the overall value. The explanation is clear and concise, making it easy to grasp the fundamental principle of dynamic programming in MDPs.This segment provides a practical application of the Bellman equation using the "student MDP" example. It demonstrates how to verify the calculated value function for a specific state by unrolling the look-ahead process and summing the expected rewards. This step-by-step calculation clarifies how the Bellman equation works in practice, making the abstract concept more concrete and understandable. This segment defines what constitutes an optimal policy in an MDP and introduces the concept of a partial ordering over policies. It explains that one policy is considered better than another if its value function is greater than or equal to the other policy's value function in all states. This provides a formal framework for comparing and evaluating different policies within the MDP context.This segment explains a crucial theorem stating that for any Markov Decision Process (MDP), there exists at least one optimal policy that is superior to or equal to all other policies. It highlights the significance of this finding, emphasizing that there's always a best way to act within an MDP to maximize rewards, and clarifies that while multiple optimal policies might exist, they all yield the same maximum reward. This segment introduces the Bellman optimality equation for the optimal value function (V*), explaining its core concept through a one-step lookahead approach. It clarifies how the optimal value of a state is determined by maximizing over the Q-values of all possible actions from that state, providing a clear and concise explanation of this fundamental equation in reinforcement learning.This segment delves into the Bellman optimality equation for the optimal action-value function (Q*), using a two-step lookahead. It explains how the optimal action value is calculated by considering the immediate reward and averaging over the possible outcomes of the environment's actions (the "wind" in the example), then maximizing over the agent's subsequent actions. This provides a comprehensive understanding of how this equation recursively relates Q* to itself. This segment describes how to find the optimal policy once the optimal action-value function (Q*) is known. It explains that selecting the action that maximizes Q* in each state provides a deterministic optimal policy, which guarantees the maximum possible reward. The explanation connects the theoretical concept to a practical approach for solving the MDP. This segment clarifies how to model large Markov Decision Processes (MDPs), focusing on reward function representation. It explains that the reward function, crucial for defining the problem, is typically a function of the state, provided by the environment (e.g., a game's score). The discussion highlights the challenge of designing reward functions that align with human intuition about optimal solutions, a key consideration in real-world applications.