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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6
votes
1answer
180 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?
5
votes
2answers
71 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 ...
4
votes
2answers
88 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 ...
3
votes
2answers
80 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 ...
2
votes
2answers
42 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
votes
1answer
82 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 ...
2
votes
1answer
51 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 ...
2
votes
2answers
73 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
votes
1answer
64 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 ...
1
vote
1answer
23 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
vote
1answer
19 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). ...