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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2
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1answer
88 views

How to prove the formula of eligibility traces operator in reinforcement learning?

I don't understand how the formula in the red circle is derived. The screenshot is taken from this paper
1
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1answer
52 views

How to derive matrix form of the Bellman operators?

Reading the Retrace paper (Safe and efficient off-policy reinforcement learning) I saw they often use a matrix form of the Bellman operators, for example as in the picture below. How do we derive ...
5
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2answers
143 views

What is the Bellman Equation actually telling?

What does the Bellman equation actually say? And are there many flavours of that? I get a little confused when I look for the Bellman equation, because I feel like people are telling slightly ...
1
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1answer
30 views

Equivalence of the $Q(s,a)$ and $V(s)$ under optimality conditions?

Are the state-action values and the state value function equivalent for a given policy? I would assume so as the value function is defined as $V(s)=\sum_a \pi(a|s)Q_{\pi}(s,a)$. If we are operating a ...
5
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1answer
66 views

How is the DQN loss derived from (or theoretically motivated by) the Bellman equation, and how is it related to the Q-learning update?

I'm doing a project on Reinforcement Learning. I programmed an agent that uses DDQN. There are a lot of tutorials on that, so the code implementation was not that hard. However, I have problems ...
4
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1answer
88 views

How are afterstate value functions mathematically defined?

In this answer, afterstate value functions are mentioned, and that temporal-difference (TD) and Monte Carlo (MC) methods can also use these value functions. Mathematically, how are these value ...
0
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0answers
38 views

Is using Bellman Optimality Equation to evaluate states a bad idea when episode number is low?

I am trying to build an RL agent that interacts with an environment, a 2D grid of dimensions 20*10: each (i,j) square in the grid gives out some reward to the agent when it visits that square. Each ...
0
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1answer
100 views

What is the optimal value function of the shifted version of the reward function?

Similarly to this question that I asked some time ago, what is the optimal value function of the shifted (by some constant $c$) version of some reward function? More precisely, let's assume that $r(s, ...
3
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1answer
49 views

Why doesn't value iteration use $\pi(a \mid s)$ while policy evaluation does?

I was looking at the Bellman equation, and I noticed a difference between the equations used in policy evaluation and value iteration. In policy evaluation, there was the presence of $\pi(a \mid s)$, ...
5
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2answers
128 views

Why are the Bellman operators contractions?

In these slides, it is written \begin{align} \left\|T^{\pi} V-T^{\pi} U\right\|_{\infty} & \leq \gamma\|V-U\|_{\infty} \tag{9} \label{9} \\ \|T V-T U\|_{\infty} & \leq \gamma\|V-U\|_{\infty} \...
6
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0answers
148 views

Is the Bellman equation that uses sampling weighted by the Q values (instead of max) a contraction?

It is proved that the Bellman update is a contraction (1). Here is the Bellman update that is used for Q-Learning: $$Q_{t+1}(s, a) = Q_{t}(s, a) + \alpha*(r(s, a, s') + \gamma \max_{a^*} (Q_{t}(s', ...
1
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1answer
37 views

If the transition model is available, why would we use sample-based algorithms?

Sample-based algorithms, like Monte Carlo Algorithms and TD-Learning, are often presented as useful since they do not require a transition model. Assuming I do have access to a transition model, are ...
1
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1answer
63 views

Connection between the Bellman equation for the action value function $q_\pi(s,a)$ and expressing $q_\pi(s,a) = q_\pi(s, a,v_\pi(s'))$

When deriving the Bellman equation for $q_\pi(s,a)$, we have $q_\pi(s,a) = E_\pi[G_t | S_t = s, A_t = a] = E_\pi[R_{t+1} + \gamma G_{t+1} | S_t = s, A_t = a]$ (1) This is what is confusing me, at this ...
3
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1answer
123 views

Why we don't use importance sampling in tabular Q-Learning?

Why don't we use an importance sampling ratio in Q-Learning, even though Q-Learning is an off-policy method? Importance sampling is used to calculate expectation of a random variable by using data ...
7
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2answers
151 views

Why state-action value function as an expected value of the return and state value function, does not need to follow policy?

I often see, the state-action value function is expressed as: $q_{\pi}(s,a)=\mathbb{E}_{\pi}[R_{t+1}+\gamma G_{t+1} | S_t=s, A_t = a] = \mathbb{E}[R_{t+1}+\gamma v_{\pi}(s') |S_t = s, A_t =a]$ Why ...
2
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1answer
133 views

Equation not satisfied in Policy Iteration Algorithm

In equation 4.9 of Sutton and Barto's book on page 79, we have(for policy iteration algo): $\pi ^{'}(s) = arg \max_{a}\sum_{s',r}p(s',r|s,a)[r+\gamma v_{\pi}(s')]$ where $\pi$ is the previous policy ...
4
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2answers
109 views

Why is $G_{t+1}$ is replaced with $v_*(S_{t+1})$ in the Bellman optimality equation?

In equation 3.17 of Sutton and Barto's book: $$q_*(s, a)=\mathbb{E}[R_{t+1} + \gamma v_*(S_{t+1}) \mid S_t = s, A_t = a]$$ $G_{t+1}$ here have been replaced with $v_*(S_{t+1})$, but no reason has ...
2
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1answer
103 views

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 ...
1
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1answer
34 views

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). ...
5
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2answers
322 views

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 ...
3
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3answers
145 views

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 ...
2
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2answers
50 views

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 ...
2
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1answer
253 views

What is the optimal value function of the scaled version of the reward function?

Consider the reward function $r(s, a)$ with optimal state-action value function $q_*(s, a)$. What would be the optimal state-action value function of $c r(s, a)$, for $c \in \mathbb{R}$? Would it be $...
16
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1answer
5k views

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 ...
3
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2answers
455 views

Apart from the state and state-action value functions, what are other examples of value functions used in RL?

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-...
3
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1answer
102 views

In reinforcement learning, does the optimal value correspond to performing the best action in a given state?

I am confused about the definition of the optimal value ($V^*$) and optimal action-value (Q*) in reinforcement learning, so I need some clarification, because some blogs I read on Medium and GitHub ...
6
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1answer
195 views

Why do Bellman equations indirectly create a policy?

I was watching a lecture on policy gradients and Bellman equations. And they say that a Bellman equation indirectly creates a policy, while the policy gradient directly learns a policy. Why is this?