LRL-8: Function Approximation - Scaling Up RL

Let's move forward to Chapter 8: Function Approximation - Scaling Up RL. We've learned powerful RL algorithms like TD Learning and Monte Carlo methods. However, these methods, in their basic form, rely on storing value functions (V or Q) in tables. This works well for small, discrete state and action spaces. But what happens when we face the real world, where state spaces can be enormous or even continuous? This chapter introduces Function Approximation as the key to scaling up RL to handle complex problems.

Chapter 8: Function Approximation - Scaling Up RL

In this chapter, we will address a critical challenge in Reinforcement Learning: scaling up to large and continuous state spaces. The tabular methods we've discussed so far (like Q-tables in Q-Learning and SARSA) become impractical when the number of states is very large or infinite. Function Approximation provides a way to generalize from a limited number of experiences to a much larger (or continuous) state space. It's the foundation for applying Reinforcement Learning to complex, real-world problems and leads us into the exciting domain of Deep Reinforcement Learning.

1. The Problem of Large State Spaces

• Curse of Dimensionality: In many realistic RL problems, the state space can be extremely large. Consider these examples:

  • Game of Go: The number of possible board positions in Go is astronomical (larger than the number of atoms in the observable universe!). Storing a Q-value for every possible board state in a table is completely infeasible.
  • Autonomous Driving: The state might include sensor data like camera images, lidar scans, GPS coordinates, speed, and the positions of other vehicles. The combination of these variables creates a continuous and high-dimensional state space.
  • Robotics: The state of a robot arm might be defined by the angles of its joints, the position of objects in its environment, and sensor readings. Again, this forms a continuous, high-dimensional state space.

• Tabular Methods Inefficiency: Tabular methods like Q-Learning and SARSA store value functions (V or Q) in tables, where each entry corresponds to a specific state (or state-action pair). For large state spaces:

  • Memory Requirements: The table size becomes prohibitively large, exceeding available memory.
  • Computational Cost: Updating and looking up values in such large tables becomes computationally expensive.
  • Lack of Generalization: Tabular methods treat each state (or state-action pair) as independent. Learning about one state provides no information about similar states. This lack of generalization is a major limitation. If you only visit a tiny fraction of the state space, you'll only learn values for those visited states, and performance in unseen states will be poor.

(Diagram: Illustrating the issue of large state space - a vast space with only a tiny portion explored by tabular methods)

+-----------------------------------------------------------------------+
|                                                                       |
|  Vast State Space                                                     |
|                                                                       |
|  +-------------------+                                              |
|  | Explored States   |  <- Tabular methods only learn about these states |
|  +-------------------+                                              |
|                                                                       |
|  Unexplored States (Vast majority!)                                   |
|                                                                       |
+-----------------------------------------------------------------------+

2. Introduction to Function Approximation: Generalizing from Limited Experience

• The Idea: Instead of storing value functions in tables, we can approximate them using parameterized functions. We represent the value function (e.g., Q-function) as a function Q̂(s, a, w), where:

  • s is the state.
  • a is the action.
  • w is a vector of parameters (weights).

  The function Q̂(s, a, w) takes a state s and action a as input and outputs an approximation of the true Q-value Q(s, a) (or Q^π(s, a)). We adjust the parameters w* to improve the approximation.

(Diagram: Function Approximation - mapping state-action to approximate Q-value via parameters)

(State s, Action a)  ----->  Function Approximator Q̂(s, a, w)  ----->  Approximate Q-value
                                      (Parameterized by w)

• Generalization: The key benefit of function approximation is generalization. By using a function, we can generalize from a limited number of experienced states to unseen states. If two states s₁ and s₂ are similar in some sense (e.g., have similar features), and our function approximator captures this similarity, then learning about Q-values for s₁ will also improve our estimate of Q-values for s₂, even if we haven't explicitly visited s₂*.

• Types of Function Approximators: We can use various types of functions for approximation, including:

  • Linear Function Approximation: Using linear combinations of features of states and actions.
  • Polynomials, Fourier Basis: Other types of basis functions.
  • Decision Trees, Nearest Neighbors: Non-linear methods.
  • Neural Networks (Deep Learning): Powerful non-linear function approximators, especially effective for high-dimensional inputs like images and raw sensor data.

3. Value Function Approximation: V̂(s, w) and Q̂(s, a, w)

• Approximating State Value Function V(s): We can approximate the state value function V^π(s) (or V(s)) as V̂(s, w). The function V̂ takes a state s and parameter vector w as input and outputs an approximate value for state s*.

  • V^π(s) ≈ V̂(s, w)

• Approximating Action Value Function Q(s, a): Similarly, we can approximate the action value function Q^π(s, a) (or Q(s, a)) as Q̂(s, a, w). The function Q̂ takes a state s, action a, and parameter vector w as input and outputs an approximate Q-value for the state-action pair (s, a)*.

  • Q^π(s, a) ≈ Q̂(s, a, w)

• Feature Vectors: Often, we don't directly use the raw state s and action a as input to the function approximator. Instead, we use feature vectors that represent the relevant characteristics of the state and action. Let φ(s) be a feature vector for state s, and φ(s, a) be a feature vector for state-action pair (s, a).

  • Linear Function Approximation (Example):

    • V̂(s, w) = w^T φ(s) = ∑_i=1ⁿ w_i φ_i(s) (Linear state value function)
    • Q̂(s, a, w) = w^T φ(s, a) = ∑_i=1ⁿ w_i φ_i(s, a) (Linear action value function)
    • Here, φ(s) or φ(s, a) are feature vectors of length n, and w is a weight vector of length n. The approximation is a linear combination of features.

  • Neural Networks (Example):

    • V̂(s, w) = NN(s; w) (State value function approximated by a Neural Network with weights w and state s as input)
    • Q̂(s, a, w) = NN(s, a; w) (Action value function approximated by a Neural Network, taking state s and action a as input, with weights w)
    • Neural networks can learn complex, non-linear relationships between inputs and outputs and are very powerful function approximators.

(Diagram: Linear Function Approximation and Neural Network Function Approximation)

Linear Function Approximation:
State s --> Feature Vector φ(s) --> Linear Combination w<sup>T</sup>φ(s) --> Approximate V(s)

Neural Network Function Approximation:
State s --> Neural Network NN(s; w) --> Approximate V(s)
State s, Action a --> Neural Network NN(s, a; w) --> Approximate Q(s, a)

4. Training Function Approximators: Stochastic Gradient Descent (SGD)

How do we learn the parameters w of our function approximator? We use methods based on Stochastic Gradient Descent (SGD).

• Objective: Minimize Error: We want to adjust the parameters w to minimize the error between our approximate value function Q̂(s, a, w) and the "true" value function (or a target value).

• Loss Function (e.g., Mean Squared Error - MSE): We define a loss function that measures this error. For example, we can use the Mean Squared Error (MSE) loss. For TD learning, a common target for Q-learning is the TD target: y = R + γ max_a' Q̂(S', a', w). The loss function for a single transition (S, A, R, S') can be:

  • L(w) = (y - Q̂(S, A, w))² = (R + γ max_a' Q̂(S', a', w) - Q̂(S, A, w))²

  (Equation: MSE Loss for Q-Learning with Function Approximation)

• Stochastic Gradient Descent (SGD): SGD is an iterative optimization algorithm used to minimize a differentiable loss function. In each step, we:

  1. Sample a transition (or batch of transitions) from experience. For example, from an experience replay buffer (we'll touch upon this later, especially in Deep RL).
  2. Calculate the TD target y (e.g., for Q-Learning: y = R + γ max_a' Q̂(S', a', w)).
  3. Calculate the gradient of the loss function L(w) with respect to the parameters w.

  4. Update the parameters w in the direction that reduces the loss:

     • w ← w - α ∇_w L(w)

     (Equation: SGD Update Rule)

     • α is the step-size (learning rate).
     • ∇_w L(w) is the gradient of the loss function with respect to w.

  5. For linear function approximation and MSE loss, the gradient update has a relatively simple form. For neural networks, we use backpropagation to calculate gradients.

Q-Learning with Function Approximation Algorithm (using SGD):

Algorithm: Q-Learning with Function Approximation (using SGD)

Input: MDP environment, function approximator Q̂(s, a, w), step-size α
Output: Trained function approximator Q̂(s, a, w) (approx. optimal Q-function)

Initialize: Parameters w of function approximator Q̂(s, a, w) (e.g., randomly)

Loop (for each episode):
    Initialize S (starting state of episode)
    Loop (for each step of episode):
        Choose action A using ε-greedy policy based on Q̂(S, ., w)
        Take action A, observe reward R and next state S'
        TD Target:  y = R + γ max<sub>a'</sub> Q̂(S', a', w)
        TD Error:   δ = y - Q̂(S, A, w)
        Update parameters w using SGD to minimize loss L(w) = (y - Q̂(S, A, w))<sup>2</sup>
        S ← S'
        If S is a terminal state, break loop

Return: Trained function approximator Q̂(s, a, w)

5. Deep Reinforcement Learning: Combining Deep Learning with RL

Deep Reinforcement Learning (Deep RL) is the combination of Reinforcement Learning with Deep Learning. It uses Deep Neural Networks as function approximators in RL algorithms.

• Why Deep Neural Networks?

  • Powerful Function Approximation: Deep Neural Networks are excellent function approximators, capable of learning highly complex, non-linear functions.
  • Automatic Feature Learning: Deep networks can automatically learn relevant features from raw, high-dimensional input data (like images, audio, text), reducing the need for manual feature engineering.
  • Scalability: Deep learning techniques have shown impressive scalability and performance in various domains, including computer vision, natural language processing, and now, Reinforcement Learning.

• Deep RL Algorithms: Many Deep RL algorithms have been developed by combining deep neural networks with various RL techniques. Some prominent examples include:

  • Deep Q-Network (DQN): Combines Q-Learning with deep neural networks to approximate the Q-function. DQN was famously used to achieve superhuman performance in Atari games. Key techniques in DQN include experience replay (storing past transitions in a buffer and sampling from it for training) and target networks (using a separate, slowly updated network for calculating TD targets to stabilize learning).

    (Image: DQN architecture - Neural Network taking game screen as input and outputting Q-values)

  • Policy Gradient Methods (e.g., REINFORCE, Actor-Critic methods like A2C, A3C, PPO, DDPG): Instead of learning value functions, policy gradient methods directly learn the policy π(a|s) using neural networks. They optimize the policy to maximize expected reward by using gradient ascent on the policy parameters.

    (Diagram: Actor-Critic architecture with Actor (policy network) and Critic (value network))

• Deep RL Successes: Deep RL has achieved remarkable successes in various domains:

  • Game Playing: Superhuman performance in Atari games, Go (AlphaGo), Chess, complex video games (StarCraft II, Dota 2).
  • Robotics: Learning complex robot control tasks, manipulation, navigation.
  • Autonomous Driving: Research in using Deep RL for autonomous driving decision-making.
  • Other Applications: Healthcare, finance, recommender systems, etc.

6. Benefits of Function Approximation (Summary)

• Generalization: Learns to generalize from experienced states to unseen states.
• Handling Large/Continuous State Spaces: Makes RL applicable to problems with vast or infinite state spaces.
• Feature Extraction (especially with Deep Learning): Automatic learning of relevant features from raw data.
• Scalability: Enables RL to tackle complex, real-world problems.

7. Challenges of Function Approximation

While function approximation is essential for scaling up RL, it also introduces new challenges:

• Instability and Divergence: With function approximation, especially non-linear function approximation like neural networks, RL algorithms can become unstable and even diverge (value function estimates can oscillate or grow infinitely). This is known as the "deadly triad" of function approximation, bootstrapping, and off-policy learning. Techniques like experience replay and target networks in DQN help to mitigate instability.
• Convergence Issues: Convergence to the optimal policy or value function is not always guaranteed with function approximation, especially with non-linear functions and off-policy methods. Theoretical guarantees become more complex.
• Parameter Tuning: Function approximation introduces hyperparameters (e.g., network architecture, learning rates, regularization) that need to be tuned, which can be challenging.

What's Next?

We have now reached a crucial point in our Reinforcement Learning journey. We've moved from basic tabular methods to the powerful concept of function approximation, especially with Deep Reinforcement Learning. This opens up the door to applying RL to a vast array of complex, real-world problems.

In the remaining chapters, we will briefly touch upon some advanced topics and real-world applications of Reinforcement Learning, including:

• Chapter 9: Exploration vs. Exploitation - The Dilemma of Learning (Revisited): We'll delve deeper into the exploration-exploitation trade-off and discuss more advanced exploration strategies.
• Chapter 10: Applications and Future of Reinforcement Learning: We'll explore real-world applications and discuss the exciting future directions of RL research.

Get ready to explore the fascinating world of advanced RL topics and see how these techniques are being used to solve real-world problems!