Answers.Body: <p>The computational complexity of Generalized Policy Iteration (GPI) in reinforcement learning depends on multiple factors, such as the number of states <span class="math-container">$S$</span>, actions <span class="math-container">$A$</span> and discount factor <span class="math-container">$γ$</span>. In iterative methods like value iteration or policy update, evaluating the value function for each state involves updating based on transitions to all possible next states which has a complexity of approximately <span class="math-container">$O(S^2A)$</span> for a single sweep of Bellman updates. And value iteration typically converges in <span class="math-container">$O(\log(\frac{1}{ 
  1−γ}))$</span> iterations since the error in the value function due to the discount factor decreases exponentially with the number of iterations, the total dominant complexity is around <span class="math-container">$O(S^2A\log(\frac{1}{ 
  1−γ}))$</span> where the complexity of policy improvement can be ignored comparatively.</p>
  <p>Convergence of GPI ultimately lies in the contraction property of the Bellman optimality operator. When the state or action space is infinite, the value functions must be approximated and the function approximation may not capture the true value perfectly. This can lead to accumulated errors during each iteration of the Bellman update and can prevent convergence to the optimal solution.</p>
  
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Answers.CreationDate: 2024-10-16T07:30:58.320Z
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Body: <p>I’m currently reading &quot;Reinforcement Learning: An Introduction&quot; by Sutton and Barto, and I have a question regarding the computational complexity of Generalized Policy Iteration (GPI) in reinforcement learning.</p>
  <p>Is there an asymptotic bound on the number of computations required to reach the optimal value function and policy? Specifically, what factors does this bound depend on, such as the number of states, actions, or transitions between states?</p>
  <p>Additionally, I understand that GPI is designed to converge to the optimal policy and value function over time. However, is it always guaranteed that GPI will converge to the optimal solution, or are there cases where convergence might fail?</p>
  
Tags: <reinforcement-learning><optimization>
Title: What Are the Computational Complexity Bounds for Dynamic Programming in Reinforcement Learning?
UserDisplayName: Cesar Ruiz
ThreadText: # What Are the Computational Complexity Bounds for Dynamic Programming in Reinforcement Learning?
  
  (Asked by: username_0 on 2024-10-16)
  
  I’m currently reading "Reinforcement Learning: An Introduction" by Sutton and Barto, and I have a question regarding the computational complexity of Generalized Policy Iteration (GPI) in reinforcement learning.
  
  Is there an asymptotic bound on the number of computations required to reach the optimal value function and policy? Specifically, what factors does this bound depend on, such as the number of states, actions, or transitions between states?
  
  Additionally, I understand that GPI is designed to converge to the optimal policy and value function over time. However, is it always guaranteed that GPI will converge to the optimal solution, or are there cases where convergence might fail?
  
  **Tags:** reinforcement-learning, optimization
  
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  ## 1 answers
  
  ### Answer by username_1 on 2024-10-16. Score: 3 | Accepted answer
  
  The computational complexity of Generalized Policy Iteration (GPI) in reinforcement learning depends on multiple factors, such as the number of states $S$, actions $A$ and discount factor $γ$. In iterative methods like value iteration or policy update, evaluating the value function for each state involves updating based on transitions to all possible next states which has a complexity of approximately $O(S^2A)$ for a single sweep of Bellman updates. And value iteration typically converges in $O(\log(\frac{1}{
  1−γ}))$ iterations since the error in the value function due to the discount factor decreases exponentially with the number of iterations, the total dominant complexity is around $O(S^2A\log(\frac{1}{
  1−γ}))$ where the complexity of policy improvement can be ignored comparatively.
  
  Convergence of GPI ultimately lies in the contraction property of the Bellman optimality operator. When the state or action space is infinite, the value functions must be approximated and the function approximation may not capture the true value perfectly. This can lead to accumulated errors during each iteration of the Bellman update and can prevent convergence to the optimal solution.
  
  
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