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Discretization is a process of converting continuous state or action spaces into discrete ones. In RL, agents often interact with environments where the state and action spaces are continuous. However, many traditional RL algorithms, especially those based on discrete spaces like Q-learning, are designed to work with discrete state and action spaces. Discretization becomes necessary when applying these algorithms to environments with continuous spaces. The idea is to divide the continuous space into a finite number of discrete bins or intervals. This allows the agent to reason about the environment in a more tractable manner. 

Some key points to consider when discretizing are: 

i) State Space Discretization: 

            For continuous state spaces, discretization involves dividing the range of each state variable into a finite number of bins. The resolution of the discretization (i.e., the number of bins) is a crucial parameter. 

            Too few bins might result in loss of important information, while too many bins can lead to increased computation and memory requirements. 

ii) Action Space Discretization: 

            Similar to state space, continuous action spaces need to be discretized for algorithms that operate on discrete action spaces. 

            The resolution of the action space discretization is important. Too coarse a discretization may lead to suboptimal policies, while too fine a discretization can be computationally expensive. 

iii) Challenges and Trade-offs: 

            Discretization introduces a form of approximation, and the choice of discretization can significantly impact the performance of the RL algorithm. 

            A challenge is finding an appropriate discretization that balances the need for computational efficiency with the preservation of important information from the continuous space. 

iv) Dynamic Discretization: 

            In some cases, it might be beneficial to use dynamic discretization, where the resolution of the discretization changes during learning. This allows the agent to adapt to the changing requirements of the task. 

v) Function Approximation:

           Discretization may not be suitable for all problems, especially when a high level of precision in representing states or actions is required. 

           In such cases, function approximation techniques, like using neural networks, can be employed to handle continuous spaces directly without discretization.

While discretization can be a useful technique in certain situations, it comes with several drawbacks and challenges in the context of reinforcement learning:

i)  Loss of Precision:

          Discretization involves dividing a continuous space into discrete bins. This can result in a loss of precision, especially if the original space is fine-grained. The coarseness of the discretization may lead to inaccuracies in representing states and actions.

ii)  Curse of Dimensionality:

          Discretization becomes challenging as the dimensionality of the state or action space increases. The number of bins required grows exponentially with the number of dimensions, leading to a phenomenon known as the curse of dimensionality. This can make learning and exploration in high-dimensional spaces computationally expensive.

Assume we have a continuous state space S in d dimensions, where each dimension i has a range [a_i, b_i]. The discretization involves dividing each dimension into n bins. The total number of bins in the discretized space is N=n^d. 

Then, the volume of the original continuous space is given by the product of the ranges in each dimension,

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The volume of each bin in the discretized space is given by,

 Then, the total number of bi   N is related to the volume of the original space and the volume of each bin,

The number of bins N) grows exponentially with the dimensionality (d) due to the multiplication of terms. This exponential growth is the essence of the curse of dimensionality.  The larger the dimensionality, the more bins are needed to adequately cover the space, making the discretization computationally expensive and leading to challenges in learning and exploration. The curse of dimensionality highlights the inherent difficulties in handling high-dimensional spaces, and it motivates the exploration of alternative methods, such as function approximation or techniques designed for continuous spaces, to address these challenges.

iv)  Increased State or Action Space Size:

          Discretization can lead to a significant increase in the size of the state or action space, especially if a fine discretization is used. This larger space can make learning more difficult and can require a more extensive amount of data and computation.

v) Non-uniformity in Density:

          Uniform discretization assumes that the bins are distributed evenly across the space. However, in practice, the density of states or actions may vary, and a uniform discretization may not capture the underlying structure effectively.

vi)  Difficulty in Handling Continuous Dynamics:

          In environments with continuous dynamics, discretization may not capture the smooth transitions between states. This can be problematic when the transition dynamics are crucial for learning optimal policies.

vii)  Sensitivity to Discretization Parameters:

          The performance of the RL algorithm can be sensitive to the choice of discretization parameters, such as the number of bins or the size of bins. Optimal parameters may vary across different problems, and selecting them can be a challenging task.

viii)  Function Approximation Challenges:

          Discretization may not be well-suited for problems where a high level of function approximation is required. In such cases, more advanced techniques like deep reinforcement learning with neural networks may be more effective in handling continuous state and action spaces.

ix)  Limited Generalization:

          Discretization may lead to a lack of generalization, as the learned policy or value function may not generalize well to unseen states or actions that fall outside the discretized representation.

Figure 3672 shows an example of discretizations in reinforcement learning.   

         
          Figure 3672. Example of discretizations in reinforcement learning (code).  

In general, the number of dimensions that can be effectively handled in practice depends on various factors, including the specific algorithms used, the amount of available data, computational resources, and the nature of the problem. In recent years, there have been significant advances in reinforcement learning (RL), particularly with the advent of deep reinforcement learning (DRL) techniques, which can handle higher-dimensional state and action spaces more effectively. While some techniques can enable RL to handle higher-dimensional spaces, in practice,   it is  still challenging to handle if the dimension  is 5 or 6 or higher.      In general, it is rare to use discretization for problems that are 8 or higher dimensional.           

When the  dimension is high, then  it is better to approximate V* directly without using discretization (see page3910).

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