Structure and Detailed Explanation of Inverse Reinforcement Learning (IRL)

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1. Core Concept of IRL

Inverse Reinforcement Learning (IRL) aims to: 1. Infer the reward function \( R(s) \) or \( R(s, a) \) from expert behavior, instead of manually defining rewards. 2. Enable the agent to obtain reasonable reward signals during learning, thereby learning a policy that mimics expert behavior.

In standard Reinforcement Learning (RL), we have a predefined reward function \( R(s, a) \) to train the agent. However, IRL works in reverse, inferring the implicit reward function \( R^*(s, a) \) given expert trajectories \( \tau_{\text{expert}} \):

\[ R^* = \arg\max_R \sum_{\tau_{\text{expert}}} P(\tau | R) \]

The agent then uses RL to optimize a policy based on the learned reward function.

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2. Overview of IRL Structure

IRL consists of four key modules: 1. Expert Trajectory Collection 2. Reward Function Learning 3. Policy Learning via RL 4. Optimization and Evaluation

The IRL structure can be visualized as follows:

            +------------------------------+
            | Expert Demonstrations (τ)   |
            +--------------+--------------+
                           |
                           v
   +-------------------+   (1) Extract Features
   | Feature Extraction | -----------------------> Features ϕ(s, a)
   +-------------------+
                           |
                           v
  +---------------------------------------------------+
  | Reward Function Learning (e.g., MaxEnt IRL, GAIL) |
  | - Train a model to approximate R(s, a)           |
  +---------------------------------------------------+
                           |
                           v
 +-------------------------------------------------+
 | Policy Learning (RL algorithm, e.g., PPO, DDPG) |
 | - Train policy π(s) to maximize learned R(s, a) |
 +-------------------------------------------------+
                           |
                           v
 +------------------------------+
 | Policy Evaluation & Refinement |
 +------------------------------+

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3. Detailed IRL Process

Step 1: Collect Expert Trajectories

First, we collect a set of expert demonstration trajectories: $$ \tau_{\text{expert}} = {(s_1, a_1), (s_2, a_2), \dots, (s_T, a_T)} $$ - These trajectories represent successful execution paths, i.e., how the expert selects actions \(a_t\) in different states \(s_t\).

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Step 2: Compute State-Action Features

We define state-action features: $$ \phi(s, a) = [\phi_1(s, a), \phi_2(s, a), ..., \phi_n(s, a)] $$ where \(\phi(s, a)\) is a high-dimensional feature vector capturing important information about state \(s\) and action \(a\).

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Step 3: Train the Reward Function

Common IRL methods include:

Method 1: Linear IRL

\[ R(s, a) = w^T \phi(s, a) \]

Method 2: Maximum Entropy IRL (MaxEnt IRL)

\[ P(\tau | R) = \frac{\exp(\sum_t R(s_t, a_t))}{Z} \]

Method 3: Generative Adversarial IRL (GAIL)

\[ R(s, a) = -\log(1 - D(s, a)) \]

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4. Advantages of IRL

✅ No need for manually designed reward functions
✅ Can learn complex behavior patterns
✅ Can generalize to different task lengths

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