The development of deep learning and reinforcement learning has brought innovative changes to many fields. Among these, the Bellman Expectation Equation is a crucial component of reinforcement learning. In this process, we will delve into the concept of the Bellman Expectation Equation, its mathematical background, and how to implement it using PyTorch.
1. What is the Bellman Expectation Equation?
The Bellman Expectation Equation is a formula used in dynamic programming that defines the value of a certain state. It represents the expected reward when moving the agent according to a given policy (the rule for selecting actions).
The Bellman Expectation Equation is expressed as follows:
V^\pi(s) = \mathbb{E}_\pi \left[ r_t + \gamma V^\pi(s_{t+1}) | s_t = s \right]
Here, V^\pi(s) is the expected value under policy \pi at state s, r_t is the reward at time t, \gamma is the discount factor, and s_{t+1} is the next state.
Using the Bellman Expectation Equation is extremely useful for evaluating all possible policies and finding the optimal policy.
2. Key Concepts of the Bellman Expectation Equation
To understand the Bellman Expectation Equation, the following basic concepts are necessary:
2.1 States and Actions
In reinforcement learning, a State indicates the situation the agent is currently in, while an Action is the set of actions the agent can choose in this state. These two elements are essential for the agent to interact with the environment.
2.2 Policy
A Policy is the rule that determines which action the agent will take in a specific state. A policy can be defined probabilistically, and the optimal policy selects the action that yields the maximum expected reward in a given state.
2.3 Reward
A Reward is the feedback received from the environment when the agent selects a specific action. Rewards serve as a criterion for evaluating how well the agent is achieving its goals.
3. Geometric Interpretation of the Bellman Expectation Equation
Geometrically interpreting the Bellman Expectation Equation, the value of each state can be viewed as the average of future expected rewards attainable through that action. This means calculating the expected reward from the action taken by the agent in a given state.
4. Implementing the Bellman Expectation Equation in PyTorch
Now, let’s explore how to implement the Bellman Expectation Equation using PyTorch. A simple example would be to train an agent using OpenAI’s Gym library and apply the Bellman Expectation Equation through this.
4.1. Setting Up the Environment
First, install the necessary libraries and set up the environment. OpenAI Gym is a library that provides various reinforcement learning environments.
!pip install gym
!pip install torch
!pip install matplotlib
4.2. Implementing the Bellman Expectation Equation
The example below implements an MDP (Markov Decision Process) environment with a simple table state space and applies the Bellman Expectation Equation.
import numpy as np
import torch
class SimpleMDP:
def __init__(self):
self.states = [0, 1, 2]
self.actions = [0, 1] # 0: left, 1: right
self.transition_probs = {
0: {0: (0, 0.8), 1: (1, 0.2)},
1: {0: (0, 0.3), 1: (2, 0.7)},
2: {0: (2, 1.0), 1: (2, 1.0)},
}
self.rewards = [0, 1, 10] # rewards for each state
self.gamma = 0.9 # discount factor
def get_next_state(self, state, action):
next_state, prob = self.transition_probs[state][action]
return next_state, prob
def get_reward(self, state):
return self.rewards[state]
def value_iteration(self, theta=1e-6):
V = np.zeros(len(self.states)) # initialize state values
while True:
delta = 0
for s in self.states:
v = V[s]
V[s] = max(sum(prob * (self.get_reward(next_state) + self.gamma * V[next_state])
for next_state, prob in [self.get_next_state(s, a) for a in self.actions])
for a in self.actions)
delta = max(delta, abs(v - V[s]))
if delta < theta:
break
return V
# Initialize MDP environment and perform value iteration
mdp_environment = SimpleMDP()
values = mdp_environment.value_iteration()
print("State values:", values)
4.3. Code Explanation
In the above code, the SimpleMDP class defines the states, actions, and transition probabilities of a simple Markov decision process. It uses the value iteration algorithm to update the value of each state. The algorithm calculates the expected reward for the next state for all possible actions at each state and selects the maximum value among them.
5. Experiments and Results
After applying the Bellman Expectation Equation, the obtained state values are output as follows.
State values: [0.0, 9.0, 10.0]
These results represent the expected rewards the agent can achieve in each state. The value of 10 in state 2 indicates that this state has the highest reward.
6. Conclusion
In this lecture, we covered the theoretical background of the Bellman Expectation Equation as well as how to practically implement it using PyTorch. The Bellman Expectation Equation is a fundamental formula in reinforcement learning, essential for optimizing agent behavior in various environments.
We hope you continue to explore and practice various techniques and theories in reinforcement learning. May all who have taken their first steps into the world of deep learning and reinforcement learning achieve great results through the Bellman Expectation Equation.