TD learning

Introduction to Temporal Difference (TD) Learning

Temporal Difference (TD) learning is a foundational concept in reinforcement learning (RL), a branch of machine learning where agents learn to make decisions by interacting with an environment. TD learning combines ideas from Monte Carlo methods and Dynamic Programming, making it a powerful approach for learning value functions directly from experience.

Key Concepts

1. Reinforcement Learning Framework

An RL problem is defined by an agent interacting with an environment over discrete time steps. At each time step ( t ):

• The agent observes a state ( s_t ),
• Takes an action ( a_t ),
• Receives a reward ( r_{t+1} ),
• And transitions to a new state ( s_{t+1} ).

2. Value Function

The value function ( V(s) ) estimates the expected future reward starting from state ( s ).
The agent’s goal is to learn a value function that helps it choose actions maximizing long-term rewards.

3. Temporal Difference Learning

Unlike Monte Carlo methods, which wait until the end of an episode to update values, TD learning updates the value estimates incrementally at each step.
The key idea is to use the difference between successive value estimates—called the TD error—to adjust ( V(s) ).

TD Learning Formula

For a given state ( s_t ), the TD update is:

[ V(s_t) \leftarrow V(s_t) + \alpha \left[ r_{t+1} + \gamma V(s_{t+1}) - V(s_t) \right] ]

Where:

• ( \alpha ) is the learning rate.
• ( \gamma ) is the discount factor (controls the importance of future rewards).
• ( r_{t+1} + \gamma V(s_{t+1}) - V(s_t) ) is the TD error.

Advantages of TD Learning

• Efficiency: Updates occur after every time step, making TD learning faster than Monte Carlo methods.
• Model-Free: TD learning does not require a model of the environment.
• Online Learning: Suitable for continuous tasks where episodes do not have a defined end.

Types of TD Methods

1. TD(0): The simplest form of TD learning where updates are based on the immediate next state.
2. TD(λ): A generalization that combines TD(0) and Monte Carlo methods, incorporating eligibility traces to balance short-term and long-term rewards.

Applications of TD Learning

• Game-playing AI (e.g., TD-Gammon for backgammon).
• Robot control.
• Financial modeling.
• Human decision-making simulations.

TD learning is central to many advanced RL algorithms, including Q-Learning and SARSA, making it a crucial stepping stone for building intelligent agents.