Markov Decision Processes

Summary: Mathematical framework for modeling sequential decision-making problems under uncertainty where an agent interacts with an environment through states, actions, and rewards. Forms the theoretical foundation for Reinforcement Learning and provides the formal structure for optimal policy computation.

Overview

A Markov Decision Process (MDP) is a discrete-time stochastic control process that satisfies the Markov property - the future state depends only on the current state and action, not on the sequence of events that led to the current state. MDPs provide the mathematical foundation for formulating sequential decision-making problems where an agent must choose actions to maximize cumulative reward over time.

The framework consists of five key components: a state space S containing all possible states, an action space A of available actions, a transition function P(s'|s,a) specifying state transition probabilities, a reward function R(s,a,s') defining immediate rewards, and a discount factor γ ∈ [0,1] for weighting future rewards. The agent's goal is to find an optimal policy π* that maps states to actions to maximize expected cumulative discounted reward.

MDPs enable the formulation of many real-world problems including robot control, game playing, resource allocation, and medical treatment planning. The framework's strength lies in its ability to capture both the sequential nature of decisions and the uncertainty inherent in many environments.

Key Details

  • Markov Property: Future states depend only on current state and action, providing memoryless transitions that enable tractable computation
  • Value Functions: State-value function V^π(s) and action-value function Q^π(s,a) measure expected cumulative reward under policy π
  • Bellman Equations: Recursive relationships V*(s) = max_a Σ P(s'|s,a)[R(s,a,s') + γV*(s')] that characterize optimal value functions
  • Policy Types: Deterministic policies π(s) → a and stochastic policies π(a|s) that specify action distributions
  • Solution Methods: Dynamic programming (value iteration, policy iteration), Monte Carlo methods, and temporal difference learning
  • Computational Complexity: Exact solution requires O(|S|²|A|) operations per iteration for value iteration
  • Discount Factor Role: γ = 0 creates myopic behavior, γ → 1 emphasizes long-term rewards, γ < 1 ensures convergence

Relationships

  • Reinforcement Learning — provides the mathematical foundation and problem formulation framework
  • State Abstraction — techniques for reducing MDP complexity by grouping states with similar value functions
  • Policy Optimization — methods for finding optimal policies in MDPs through gradient-based and other approaches
  • Q-Distance — metrics for measuring differences in action-values between states in MDP value functions
  • Decision-Relevant Concepts — interpretable features that preserve MDP decision structure through state abstraction
  • Concept-Based Models — interpretable approaches that map MDP states through human-understandable concepts
  • Approximate State Abstraction — methods for creating tractable MDP approximations while preserving optimal decision-making

Sources