Scaling Q-Learning: From Tables to Neural Networks

Last updated: November 25, 2024

1. Introduction

Previously, we discussed the fundamentals of Q-learning, including its core parameters—discount factor, learning rate, and exploration strategies—and its off-policy learning capability. While effective for small-scale problems, traditional Q-learning struggles with scalability in complex environments due to its reliance on tabular methods for storing Q-values for every state-action pair. This becomes impractical as the state space grows exponentially.

2. Limitations of Tabular Q-Learning

Scalability:
- In simple environments like Grid World, storing a Q-table is feasible.
- For larger problems like Tetris (~10⁶⁰ states) or Atari games (~10²⁰⁰ to 10³⁰⁰ states), tabular methods become computationally prohibitive.
Continuous State Spaces:
- Robotics environments often involve continuous state spaces (e.g., Humanoid control with ~10¹⁰⁰ states).
- Discretizing these spaces loses essential detail, undermining performance.
Memory Constraints:
- The memory required to store Q-values grows exponentially with the number of states, making tabular approaches impractical for real-world applications.

3. Transition to Approximate Q-Learning

To overcome the limitations of tabular Q-learning in large or continuous environments, Approximate Q-Learning replaces Q-tables with parameterized functions. These functions model the relationship between state-action pairs and their Q-values, allowing agents to generalize from limited experiences to unseen states. This makes learning feasible and efficient even in high-dimensional state spaces.

Instead of maintaining a Q-value table, a parameterized function estimates $Q(s, a)$ for any state-action pair. The function uses parameters $\theta$, which are adjusted during training to minimize the error between estimated and true Q-values.

Common Parameterization Techniques

Linear Functions:
- Early approaches modeled the Q-function as a weighted sum of state features.
- Features represent specific characteristics of the state. For example, in Tetris:
  - Features might include column heights, the number of gaps, or the lines cleared.
  - Q-values are computed as:
    $$Q(s, a) = \sum_{i} \theta_i \cdot \phi_i(s, a)$$, where $\phi_i(s, a)$ are the features, and $\theta_i$ are the corresponding weights.
- While computationally simple, linear functions lack the expressive power to capture complex relationships in high-dimensional spaces.
Neural Networks:
- Modern approaches use neural networks to parameterize $Q(s, a)$.
- The network takes state (and sometimes action) as input, processes it through hidden layers, and outputs a Q-value.
- By adjusting network weights, the model learns to approximate Q-values even for states and actions it has never encountered.
- Neural networks excel in handling non-linear and complex relationships, making them suitable for tasks with high-dimensional inputs, such as image-based state representations in games or robotics.

4. Neural Network-Based Q-Learning: Core Concepts

Target Q-Value Calculation

At each step, the agent computes a target Q-value:
$$Q_{\text{target}} = r + \gamma \max_{a'} Q(s', a'; \theta_{\text{target}})$$
- $r$: immediate reward from the environment.
- $\gamma$: discount factor, balancing immediate and future rewards.
- $s'$: the next state after taking action $a$ in state $s$.

Loss Function

The network updates its weights by minimizing the difference between the predicted Q-value and the target Q-value:
$$
\mathcal{L}(\theta) = \mathbb{E}\left[(Q_{\text{target}} - Q(s, a; \theta))^2\right]
$$
This quadratic loss ensures the network learns to align its predictions with the target values.

Gradient-Based Optimization

Using stochastic gradient descent (SGD) or its variants, the network iteratively updates its weights $\theta$:
$$ \theta \gets \theta - \alpha \nabla_\theta \mathcal{L}(\theta) $$
- $\alpha$: learning rate, determining the step size for updates.a
- $\nabla_\theta \mathcal{L}(\theta)$: gradient of the loss function with respect to $\theta$.

Scalability

Neural networks generalize Q-values across similar states, avoiding the need to explicitly store Q-values for all state-action pairs. This enables agents to handle high-dimensional and continuous spaces.

4a. Process Overview for Neural Network Q-Learning

Input: The current state $s$ and action $a$ are fed into the neural network.
Target: Compute the target Q-value using the reward $r$ and the estimated future Q-value $\max_{a'} Q(s', a'; \theta_{\text{target}})$.
Loss: Calculate the loss as the squared difference between the predicted Q-value and the target Q-value.
Optimization: Update network weights { $\theta}$ using gradient descent to minimize the loss and improve future predictions.

4b. Advantages of Approximate Q-Learning

Memory Efficiency: Avoids storing explicit Q-values for all state-action pairs.
Generalization: Learns patterns that can be applied to unseen states, essential for large and continuous spaces.
Expressive Power: Neural networks capture complex relationships, enabling effective learning in challenging environments.

By transitioning from tabular methods to parameterized functions, Approximate Q-Learning extends reinforcement learning to domains previously inaccessible due to scalability constraints. In particular, neural networks have revolutionized this approach, laying the groundwork for advanced methods like Deep Q-Learning, where deep architectures tackle even more complex tasks.

5. Summary

We examined traditional Q-learning and its scalability challenges in large or continuous environments. Approximate Q-learning, particularly with neural networks, addresses these limitations by enabling generalization across states, reducing memory demands, and extending applicability to complex environments. This foundation leads naturally to Deep Q-Learning, where deep neural networks tackle even more advanced tasks, such as image-based environments.