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## Hot answers tagged return

3

In general, $\mathbb{E}_\pi[G_{t:t+n}|S_t = s] \neq v_\pi(s)$. $v_\pi(s)$ is defined as $\mathbb{E}_\pi[\sum_{k=0}^{\infty} \gamma^k R_{t+k+1} | S_t = s]$, so you should be able to see why the two are not equal when the LHS is an expectation of the $n$th step return. They would only be equal as $n \rightarrow \infty$.

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$T = \infty$ and $\gamma = 1$ cannot be both true at the same time because the return defined in equation 3.11 is supposed to be a unified definition of the return for both continuing and episodic tasks. In the case of continuing tasks, $T = \infty$ and $\gamma = 1$ cannot be true at the same time, because the return may not be finite in that case (as I ...

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It wouldn't make sense to define the return as you propose, from time 0 to $t$. Once we are in a state at time $t$ we don't care what the returns have been, rather what they will be in the future, thus returns are defined as the total sum of discounted rewards from the current time step onwards. This allows the agent to make decisions about which actions to ...

3

However, from the blogs and texts I read, the equations are expressed in terms of V and NOT Q. Why is that? MC and TD are methods for associating value estimates to time step based on experienced gained in later time steps. It does not matter what kind of value estimate is being associated across time, because all value functions are expressing the same ...

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Can someone provide the reasoning behind why $G_{t+1}$ is equal to $v_*(S_{t+1})$? The two things are not usually exactly equal, because $G_{t+1}$ is a probability distribution over all possible future returns whilst $v_*(S_{t+1})$ is a probability distribution derived over all possible values of $S_{t+1}$. These will be different distributions much of the ...

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You know all the rewards. They're 5, 7, 7, 7, and 7s forever. The problem now boils down to essentially a geometric series computation. $$G_0 = R_0 + \gamma G_1$$ $$G_0 = 5 + \gamma\sum_{k=0}^\infty 7\gamma^k$$ $$G_0 = 5 + 7\gamma\sum_{k=0}^\infty\gamma^k$$ $$G_0 = 5 + \frac{7\gamma}{1-\gamma} = \frac{5 + 2\gamma}{1-\gamma}$$

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Note that for a general policy $\pi$ we have that $q_{\pi}(s,a) = \mathbb{E}_{\pi}[G_t | S_t = s, A_t = a]$, where in state $S_t$ we take action $a$ and thereafter following policy $\pi$. Note that the expectation is taken with respect to the reward transition distribution $\mathbb{P}(R_{t+1} = r, S_{t+1} = s' | A_t = a, S_t = s)$ which I will denote as $p(s'... 2 The discussion uses poor notation, there should be a time index. You obtain a list of tuples$(s_t, a_t, r_t, s_{t+1})$and then, for every visit MC, you update $$Q(s_t, a_t) = Q(s_t, a_t) + \alpha (G_t - Q(s_t, a_t))\;;$$ where$G_t = \sum_{k=0}^\infty \gamma^k r_{t+k}$, for each$t$in the episode. You can see that the returns for each time step are ... 2 Return refers to the total discounted reward, starting from the current timestep. 2 Why is the expected return in Reinforcement Learning (RL) computed as a sum of cumulative rewards? That is the definition of return. In fact when applying a discount factor this should formally be called discounted return, and not simply "return". Usually the same symbol is used for both ($R$in your case,$G$in e.g. Sutton & Barto). There ... 1 There are a few ways to resolve values of infinite sums. In this case, we can use a simple technique of self-reference to create a solvable equation. I will show how to do it for the generic case here of an MDP with same reward$r$on each timestep: $$G_t = \sum_{k=0}^{\infty} \gamma^k r$$ We can "pop off" the first item:$\$G_t = r + \sum_{k=1}^{\...

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According to my understanding, you don't use just the current behavior policy for sampling. The importance sampling ratio is calculated as the product of the probability ratios for both the target and behaviour policy throughout the trajectory. See the calculation below, where the product is happening for all the probabilities throughout the trajectories. (...

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As the accepted answer states, the return at the current timestep is equal to the sum of discounted rewards from all future timesteps until the end of the episode. In Chapter 5 of Sutton and Barto, returns must be used to estimate the state-value and action-value functions because episode lengths are unrestricted and may be greater than one. In contrast, ...

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