# Questions tagged [bellman-equations]

For questions related to the Bellman equations in the context of reinforcement learning (and other artificial intelligence subfields).

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### Proof that the Policy Iteration Converges?

Let $\mathcal{X}=:\{x_1, x_2, x_3,...,x_n\}$ be the state space. Let $\mathcal{U}:=\{u_1, u_2, u_3,...,u_m\}$ be the set of actions. Let $A^{u_1}, A^{u_2}, A^{u_3},...,A^{u_m}$ be the state transition ...
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### Bellman optimality equation for continuous variables

I'm currently studying Reinforcement Learning and I'd like to know what would be the Bellman optimality equation for action values $q_∗(s, a)$ for a MDP with continuous states and actions, written out ...
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### Is my derivation of the Bellman equation for $q_{\pi}$ in terms of $p(s'|s,a)$ and $r(s,a)$ correct?

I have done exercise 3.29 from Sutton and Barto and I'd like to check if it's correct. Here's the exercise: Rewrite the Bellman equation for the function $q_{\pi}$ in terms of the three argument ...
• 81
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### Is my derivation of the Bellman equation for $v_{\pi}$ in terms of $p(s'|s,a)$ and $r(s,a)$ correct?

I have exercise 3.29 from Sutton and Barto and I'd like to check if it's correct. Here's the exercise: Rewrite the Bellman equation for the value function $v_{\pi}$ in terms of the three argument ...
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1 vote
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### Value iteration algorithm converge to max reward with discount of 1? [duplicate]

I'm running the value iteration algorithm of a gridworld of 4*3, with two terminal nodes with -50 reward and one with +20 reward like so: ...
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### Why is the projection reasonable in C51 algorithm?

In this paper, there is an equation about the projection. The author said we project the sample Bellman update $\hat{T}Z_θ$ onto the support of $Z_θ$ Why is the original support reasonable? And why ...
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### Can we also estimate $V_{\pi}$ with SARSA?

For SARSA, I know we can estimate the action value $Q(s,a)$, and the relationship between $V(s)$ and $Q(s,a)$ is $V_{\pi}(s) = \sum_{a \in \mathcal{A}} \pi(a|s)Q_{\pi} (s,a)$. So my question is, can ...
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### What is so special about the Bellman Optimality Principle?

In the context of Decision Making and Game Theory, "Bellman's Equations and Bellman's Conditions of Optimality" are said to be some of the most important mathematical principles in this ...
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### Determining to terminate at a reward or not

I am practicing the Bellman equation on Grid world examples and in this scenario, there are numbered grid squares where the agent can choose to terminate and collect the reward equal to the amount ...
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1 vote
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### What is the Bellman equation for V(s) in the case of a deterministic environment?

I am currently trying to practice reinforcement learning for an agent on a grid. The grid is deterministic. Since the grid is deterministic, to calculate the value for each grid square from the reward ...
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### Determine Gridworld values

I am learning Reinforcement learning for games following Gridworld examples. Apologies in advance if this is a basic question, very new to reinforcement learning. I am slightly confused in scenarios ...
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### What is the difference between gradient decent in neural networks and temporal difference in reinforcement learning?

I am studying Q-learning in reinforcement learning. My question is about the Bellman equation. In Q-learning, the Bellman equation is often introduced as follows. \begin{align} Q_{new}(s,a) &= Q_{...
1 vote
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### How can we find the value function by solving a system of linear equations?

I am following the book "Reinforcement Learning: An Introduction" by Richard Sutton and Andrew Barto, and they give an example of a problem for which the value function can be computed ...
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### What do the terms 'Bellman backup' and 'Bellman error' mean?

Some RL literature use terms such as: 'Bellman backup' and 'Bellman error'. What do these terms refer to?
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### Are these two definitions of the state-action value function equivalent?

I have been reading the Sutton and Barto textbook and going through David Silvers UCL lecture videos on YouTube and have a question on the equivalence of two forms of the state-action value function ...
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### How are the Bellman optimality equations and minimax related?

Is the philosophy between Bellman equations and minimax the same? Both the algorithms look at the full horizon and take into account potential gains (Bellman) and potential losses (minimax). ...
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### What is the proof that policy evaluation converges to the optimal solution?

Although I know how the algorithm of iterative policy evaluation using dynamic programming works, I am having a hard time realizing how it actually converges. It appeals to intuition that, with each ...
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### Why can the Bellman equation be turned into an update rule?

In chapter 4.1 of Sutton's book, the Bellman equation is turned into an update rule by simply changing the indices of it. How is it mathematically justified? I didn't quite get the initiation of why ...
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### Why is there an expectation sign in the Bellman equation?

In chapter 3.5 of Sutton's book, the value function is defined as: Can someone give me some clarification about why there is the expectation sign behind the entire equation? Considering that the ...
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### What is the Bellman operator in reinforcement learning?

In mathematics, the word operator can refer to several distinct but related concepts. An operator can be defined as a function between two vector spaces, it can be defined as a function where the ...
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In reinforcement learning, we often define two functions, the state-value function $$V^\pi(s) = \mathbb{E}_{\pi} \left[\sum_{k=0}^{\infty} \gamma^{k}R_{t+k+1} \Bigg| S_t=s \right]$$ and the state-...