Overview of TD learning
TD (Temporal Difference) learning is a type of Reinforcement Learning, a method for agents to learn how to maximise rewards while interacting with their environment. TD Learning uses the difference between the actual observed reward and the predicted future reward to update the prediction of future rewards (Temporal Difference).
1. Online learning: TD learning uses the information obtained by the agent at each step of the learning process. This means that the agent learns in real-time as it explores the environment.
2. bootstrapped learning: TD learning uses future predictions to update current predictions. This approach is known as ‘bootstrapping’ and is characterised by its reliance on future predictions for learning.
The basic elements of TD learning are as follows.
- State, \(S \): indicates the situation the agent is currently in.
- Action (Action, \ A \): the action taken by the agent.
- Reward, \(R \)): feedback obtained from the environment for a particular action.
- Value Function, \(V(s) \): a prediction of the expected cumulative reward in state \(S \).
The basic update rule for TD learning is expressed as follows.
\[ V(S_t) \leftarrow V(S_t) + \alpha \left[ R_{t+1} + \gamma V(S_{t+1}) – V(S_t) \right] \]
Where:
– \( V(S_t) \) is the value of the current state \( S_t \).
– \( R_{t+1} \) is the reward obtained at the next time step \( t+1 \).
– \( \gamma \) is the discount rate, used to calculate the present value of future rewards.
– \( \alpha \) is the learning rate, which determines how sensitive the animal is to new information.
TD learning has been applied in a variety of fields, including board games such as chess and Go, robot control and financial market modelling, and the real-time adaptive capabilities and efficiency of TD learning provide powerful solutions to many real-world problems.
Algorithms related to TD learning.
The following sections describe the algorithms associated with typical TD learning.
1. TD(0): TD(0) is the most basic TD learning algorithm, which updates the value function using the prediction one step ahead. The update rule is expressed as follows.
\[ V(S_t) \leftarrow V(S_t) + \alpha \left[ R_{t+1} + \gamma V(S_{t+1}) – V(S_t) \right] \]
2. SARSA: SARSA (State-Action-Reward-State-Action) is an on-policy TD learning algorithm. The agent chooses an action according to the current policy and updates the value function based on that action. The update rules are as follows.
\[ Q(S_t, A_t) \leftarrow Q(S_t, A_t) + \alpha \left[ R_{t+1} + \gamma Q(S_{t+1}, A_{t+1}) – Q(S_t, A_t) \right] \]
For more information on SARSA, see “Overview of SARSA and its algorithms and implementation systems“.
3. Q-learning: Q-learning is an off-policy TD learning algorithm. The agent uses the largest Q-value it can obtain in the next state to update in order to find the best policy. The update rules are as follows.
\[ Q(S_t, A_t) \leftarrow Q(S_t, A_t) + \alpha \left[ R_{t+1} + \gamma \max_{a} Q(S_{t+1}, a) – Q(S_t, A_t) \right] \]
For more information on Q-learning, see “Overview of Q-learning and examples of algorithms and implementations“.
4. TD(λ): TD(λ) is an algorithm that lies between TD(0) and the Monte Carlo method, using the λ (lambda) parameter to weight and average the predictions of the different steps. It uses a concept called Eligiblity Trace.The update rule for TD(λ) is as follows.
\[ V(S_t) \leftarrow V(S_t) + \alpha \left[ R_{t+1} + \gamma V(S_{t+1}) – V(S_t) \right] e_t \]
Where \( e_t \) is the elegance trace and decays with time.
5. double Q learning: to reduce bias in Q learning, double Q learning has been proposed. It has two Q-value functions, each of which is updated alternately to prevent over-optimism. The update rules are as follows.
\[ Q_1(S_t, A_t) \leftarrow Q_1(S_t, A_t) + \alpha \left[ R_{t+1} + \gamma Q_2(S_{t+1}, \arg\max_a Q_1(S_{t+1}, a)) – Q_1(S_t, A_t) \right] \]
\[ Q_2(S_t, A_t) \leftarrow Q_2(S_t, A_t) + \alpha \left[ R_{t+1} + \gamma Q_1(S_{t+1}, \arg\max_a Q_2(S_{t+1}, a)) – Q_2(S_t, A_t) \right] \]
6. the Actor-Critic method: the Actor-Critic method is a TD learning algorithm with two networks: the policy (Actor) and the value function (Critic), where the Critic calculates the TD error and the Actor updates the policy using it. For more information, see Actor-Critic Overview, Algorithm and Implementation Examples.
7. DDPG (Deep Deterministic Policy Gradient): DDPG is an algorithm for continuous action spaces based on the Actor-Critic method, which uses deep learning to deal with large state and action spaces. For more information, see Deep Deterministic Policy Gradient (DDPG) Overview, Algorithm and Implementation Examples.
Application examples of TD learning
Temporal Difference (TD) learning has been widely applied in various fields. Typical applications of TD learning are described below.
1. game AI:
Examples: chess, Go, backgammon
TD learning is very effective in game AI. A particularly well-known case study will be TD-Gammon, developed by Gerald Teslo. This backgammon player achieves world champion-level play by repeatedly playing against itself using TD(λ) learning.
2. robot control:
Case study: robot navigation and manipulation
TD learning is used when a robot navigates an unknown environment or performs a specific task. For example, SARSA and Q-learning are used to learn the best path for the robot to reach its destination while avoiding obstacles.
3. self-driving vehicles:
Case study: optimising vehicle path planning and driving behaviour.
TD learning is also used by automated vehicles to learn optimal driving behaviour in real-time. It uses the feedback provided by the agent as it drives to learn policies to reach its destination efficiently while complying with traffic rules.
4. finance:
Case study: optimising stock trading strategies
TD learning has also been applied to optimise trading strategies in the stock market. Reinforcement learning agents use market data to learn to optimise the timing of trades and investment portfolios, e.g. using Q-learning and TD(λ) to construct risk- and return-aware trading strategies.
5. healthcare:
Case study: optimising treatment policies
TD learning is also used to optimise patient treatment strategies. Based on the patient’s health status and treatment response, an optimal treatment policy can be learnt, thereby providing an optimised treatment plan for each individual patient.
6. personalised recommendations:
Case study: film and product recommendation system
TD learning is also being used to build personalised recommendation systems based on users’ past behaviour. For example, platforms such as Netflix and Amazon use TD learning to learn which films and products to recommend next based on users’ viewing and purchase history.
7. optimising sports strategy:
Case study: sports teams’ tactical decisions
TD learning is also used by sports teams to optimise their in-game tactics. For example, TD algorithms are used to learn basketball teams to adjust their in-game strategy in real-time to maximise their chances of winning.
8. computer networks:
Case study: packet routing
TD learning is also applied to optimising packet routing in computer networks. By monitoring network traffic and learning optimal routes in real-time, communication delays are minimised and network efficiency is improved.
Examples of TD learning implementations
As an example of TD learning implementation, a simple reinforcement learning agent using the TD(0) algorithm is presented here in Python. In this example, we consider a scenario where the agent learns to reach a goal in a one-dimensional gridworld environment.
1. environment definition: first, a one-dimensional gridworld environment is defined. In this environment, the agent moves from the start position towards the goal position and is rewarded for reaching the goal.
import numpy as np
class GridWorld:
def __init__(self, size, start, goal):
self.size = size
self.start = start
self.goal = goal
self.reset()
def reset(self):
self.state = self.start
return self.state
def step(self, action):
if action == 0: # move left
next_state = max(0, self.state - 1)
elif action == 1: # move right
next_state = min(self.size - 1, self.state + 1)
reward = 1 if next_state == self.goal else 0
done = next_state == self.goal
self.state = next_state
return next_state, reward, done2. implementation of a TD(0) agent: we next implement an agent that learns a value function using the TD(0) algorithm.
class TDAgent:
def __init__(self, env, alpha=0.1, gamma=0.9):
self.env = env
self.alpha = alpha
self.gamma = gamma
self.value_function = np.zeros(env.size)
def choose_action(self):
return np.random.choice([0, 1]) # random policy: left or right
def update_value_function(self, state, reward, next_state):
td_target = reward + self.gamma * self.value_function[next_state]
td_error = td_target - self.value_function[state]
self.value_function[state] += self.alpha * td_error
def train(self, episodes):
for episode in range(episodes):
state = self.env.reset()
done = False
while not done:
action = self.choose_action()
next_state, reward, done = self.env.step(action)
self.update_value_function(state, reward, next_state)
state = next_state3. train and check the results: finally, train the agent and see how the value function is updated.
# Initialising the environment and agents
env = GridWorld(size=5, start=0, goal=4)
agent = TDAgent(env)
# training
agent.train(episodes=100)
# Confirmation of results
print("Learned value function:")
print(agent.value_function)Code execution: when the above code is executed, the agent learns a value function for reaching a goal in a one-dimensional grid-world environment. The output of the value function after training allows the agent to see how it has assessed the value of each state.
This implementation example shows how an agent learns in a simple grid-world environment using the TD(0) algorithm; it is a simple example that helps to understand the basic ideas of TD learning. More complex environments, improved policies and experimenting with different TD algorithms (e.g. SARSA and Q-learning) will enable the construction of even more powerful reinforcement learning agents.
Challenges and measures for TD learning.
Temporal Difference (TD) learning is a powerful reinforcement learning technique, but it also comes with some challenges. The main challenges of TD learning and the measures taken to address each are described below.
1. the trade-off between exploration and exploitation:
Challenge: TD learning requires agents to explore new information while utilising known information to maximise rewards. This balance is difficult to achieve.
Solution:
ε-greedy policy: a method that selects a random behaviour with ε probability and a behaviour considered optimal with 1-ε probability.
Soft-max policy: the probability of selecting an action is calculated based on the Q-value and the temperature parameter is adjusted to achieve a balance between exploration and utilisation.
UCB (Upper Confidence Bound): a method of selecting an action to explore an unknown state, taking into account the uncertainty of the state.
2. scalability in large state spaces:
Challenge: as state spaces become larger, it becomes difficult to explicitly maintain value functions for all states.
Solution:
Function approximation: using neural networks and linear function approximation to approximate state values, e.g. Deep Q-Networks (DQN).
Feature engineering: converting states into low-dimensional feature vectors to facilitate learning.
3. adaptation to non-stationary environments:
Challenge: if the environment changes over time, previously learned knowledge may become outdated.
Solution:
Adaptive learning rate: adjust sensitivity to new information by varying the learning rate over time.
Forgetting coefficients: use attenuation coefficients in the calculation of rewards and TD errors to gradually reduce the impact of old information.
4. sample efficiency:
Challenge: TD learning can require a large number of samples.
Solution:
Experience replay: improving sample efficiency by reusing previous experiences; technique used in DQN.
Model-based reinforcement learning: learns a model of the environment and uses it to generate virtual experiences.
5. trade-offs between bias and variance:
Challenge: in TD learning, it is important to balance bias (systematic error) and variance (variability in estimation) in value estimation.
Solution:
TD(λ): take an intermediate approach between TD(0) and Monte Carlo methods by adjusting the λ parameter. This allows the trade-off between bias and variance to be adjusted.
Dual Q-learning: a method that uses two Q-value functions to reduce the Q-value estimation bias.
6. choice of discount rate (γ):
Challenge: the choice of discount rate γ has a significant impact on the speed of learning convergence and the quality of the policy.
Solution:
Tuning of γ: experiment with multiple values to find the best γ.
Tuning based on characteristics of the environment: set γ based on the time span of the goal and the nature of the reward.
References and Reference Books
Details of reinforcement learning are described in “Theories and Algorithms of Various Reinforcement Learning Techniques and Their Python Implementations. Please also refer to this page.
A reference book is “Reinforcement Learning: An Introduction, Second Edition.
“Deep Reinforcement Learning with Python: With PyTorch, TensorFlow and OpenAI Gym“
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