Introduction to Markov Decision Processes

Introduction to Markov Decision Processes

• Markov Decision Processes (MDPs) are mathematical frameworks used to model decision-making in situations with uncertainty.
• They represent sequential decision-making problems where the outcome of an action depends on the current state and the action taken.
• MDPs are widely used in reinforcement learning, robotics, and other fields requiring sequential decision-making.

Key Components of an MDP

• States: A set of possible states the system can be in. Each state represents a specific configuration of the system.
• Actions: A set of actions the agent can take. The available actions may depend on the current state.
• Transition Probabilities: A probability distribution describing the likelihood of transitioning from a given state to another state when a particular action is taken. Mathematically represented as P(s' | s, a)—the probability of transitioning to state s' given the current state s and the action a.
• Rewards: A numerical value associated with each state-action pair (or sometimes specific transitions). The reward function represents the immediate benefit or cost of being in a given state and taking a specific action. Rewards are typically used to guide the agent towards desirable outcomes.
• Discount Factor (γ): A value between 0 and 1 that determines the importance of future rewards. A higher discount factor emphasizes future rewards more than immediate rewards. It discounts future rewards to make them less valuable than immediate rewards.

Defining the Problem

• The agent's goal in an MDP is to find a policy that maximizes the expected cumulative reward over time. A policy is a mapping from states to actions.
• The optimal policy specifies the best action to take in each state.

Types of Policies

• Deterministic Policy: Assigns a unique action to every state.
• Stochastic Policy: Assigns a probability distribution of actions to each state.

Reward Function

• The reward function plays a critical role in guiding the agent toward its goal.
• Rewards can be positive (for beneficial outcomes) or negative (for detrimental outcomes).
• The choice of reward function significantly impacts the optimal policy.

MDP Solution Methods

• Value Iteration: An iterative method for finding the optimal policy. It computes the optimal value function, which represents the expected cumulative reward starting from a given state.
• Policy Iteration: Another iterative method that alternates between evaluating a policy and improving it to find the optimal policy.

Applications of MDPs

• Robotics: Controlling robots in dynamic environments.
• Resource Management: Optimizing allocation of resources.
• Game Playing: Developing strategies for games involving sequential decisions.
• Finance: Managing investments and portfolios over time.
• Inventory Control: Managing inventory levels to meet demand.

Summary of Important Concepts

• Understanding the state space, action space, and transitions are important parts of MDP definition.
• An appropriate reward function is crucial to guide the system towards the desired outcome.
• Value and policy iteration are standard techniques for learning optimal policies in MDPs.