The Overestimation Problem in Q-Learning

Standard Deep Q-Networks represent a significant step forward, allowing us to apply reinforcement learning to problems with high-dimensional state spaces. However, the underlying Q-learning mechanism, particularly its update rule, carries an inherent tendency towards optimism: it can systematically overestimate action values. This isn't just a minor inaccuracy; it can negatively impact learning performance and stability. Let's examine why this happens.

Recall the standard Q-learning update rule used in tabular methods:

Q(s, a) \leftarrow Q(s, a) + \alpha [r + \gamma \max_{a'} Q(s', a') - Q(s, a)]

The critical part is the $\max_{a'} Q(s', a')$ term. This component estimates the maximum possible value achievable from the next state $s'$ . The problem arises because $Q(s', a')$ are themselves estimates, especially early in training or when using function approximators like neural networks. These estimates inevitably contain noise or errors.

Consider what happens when you take the maximum over several noisy estimates. If some estimates are randomly higher than their true values and others are randomly lower, the $\max$ operation is more likely to select one of the overestimated values. It doesn't average out the errors; it actively picks the largest value it sees, potentially amplifying positive noise. This leads to a consistent positive bias in the estimated value of the next state.

This phenomenon is known as maximization bias. Imagine you have several actions available in state $s'$ , and your current $Q$ -function estimates their values with some random error:

The true best action is Action 2 with a value of 1.5. However, due to estimation error, the agent's estimate for Action 2 is 1.9. The $\max$ operator selects this overestimated value (1.9) to use in the target calculation, leading to a positive bias compared to the true maximum value (1.5).

In Deep Q-Networks, this issue persists. The target value $y_i$ used for training the network is calculated as:

y_i = r_i + \gamma \max_{a'} Q_{target}(s'_{i}, a'; \theta^{-})

Even though we use a separate target network $Q_{target}$ with parameters $\theta^{-}$ to stabilize training, the $\max$ operation is still applied to the estimated values produced by this target network. If these target $Q$ -values are noisy or uncertain, the maximization step will continue to introduce an upward bias into the targets $y_i$ .

What are the consequences of this overestimation?

Suboptimal Policies: The agent might learn to prefer actions that lead to states where the $Q$ -values are significantly overestimated, rather than states that are truly optimal.
Slowed Convergence: The bias can make the learning process less stable and potentially slower, as the network tries to match these inflated target values.
Instability: In some cases, severe overestimation can lead to oscillations or divergence during training.

Understanding this maximization bias is important because it motivates several improvements to the basic DQN algorithm. Techniques have been developed specifically to mitigate this overestimation problem, leading to more reliable and efficient learning. One of the most direct solutions is Double DQN, which modifies how the target value is calculated.

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References

Reinforcement Learning: An Introduction, Richard S. Sutton and Andrew G. Barto, 2018 (MIT Press) - Comprehensive textbook explaining Q-learning, temporal difference learning, and the maximization bias problem, along with Double Q-learning as a solution.
Double Q-learning, Hado van Hasselt, 2010 Advances in Neural Information Processing Systems (NIPS), Vol. 23 (Neural Information Processing Systems Foundation) - Introduces Double Q-learning to mitigate maximization bias in reinforcement learning, providing theoretical analysis.
Deep Reinforcement Learning with Double Q-learning, Hado van Hasselt, Arthur Guez, and David Silver, 2016 Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence (AAAI) - Extends Double Q-learning to Deep Q-Networks (DQN), demonstrating its effectiveness in reducing overestimation and improving performance.