Questions tagged [expectation]

For questions related to the mathematical concept of "expectation" or "expected value".

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80 views

What is wrong with equation 7.3 in Sutton & Barto's book?

Equation 7.3 of Sutton Barto book: $$\text{Equation: } max_s|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi| \le \gamma^nmax_s|V_{t+n-1}(s) - v_\pi(s)| $$ $$\text{where }G_{t:t+n} = R_{t+1} + \gamma R_{t+2}...
2
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1answer
32 views

Why is the mean used to compute the expectation in the GAN loss?

From Goodfellow et al. (2014), we have the adversarial loss: $$ \min_G \, \max_D V (D, G) = \mathbb{E}_{x∼p_{data}(x)} \, [\log \, D(x)] \\ \quad\quad\quad\quad\quad\quad\quad + \, \mathbb{E}_{z∼p_z(...
2
votes
2answers
160 views

How is per-decision importance sampling derived in Sutton & Barto's book?

In per-decison importance sampling given in Sutton & Barto's book: Eq 5.12 $\rho_{t:T-1}R_{t+k} = \frac{\pi(A_{t}|S_{t})}{b(A_{t}|S_{t})}\frac{\pi(A_{t+1}|S_{t+1})}{b(A_{t+1}|S_{t+1})}\frac{\pi(...
0
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0answers
84 views

Problem in understanding equation given for convergence of TD(n) algorithm

Given equation 7.3 of Sutton and Barto's book for convergence of TD(n): $\max_s|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)| \leqslant \gamma^n \max_s|V_{t+n-1}(s) - v_\pi(s)|$ $\textbf{PROBLEM ...
3
votes
1answer
81 views

How does $\mathbb{E}$ suddenly change to $\mathbb{E}_{\pi'}$ in this equation?

In Sutton-Barto's book on page 63 (81 of the pdf): $$\mathbb{E}[R_{t+1} + \gamma v_\pi(S_{t+1}) \mid S_t=s,A_t=\pi'(s)] = \mathbb{E}_{\pi'}[R_{t+1} + \gamma v_\pi(S_{t+1}) \mid S_{t} = s]$$ How does $...
4
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2answers
107 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
95 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 ...
3
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1answer
36 views

What is meant by the expected BLEU cost when training with BLEU and SIMILE?

Recently I was reading a paper based on a new evaluation metric SIMILE. In a section, validation loss comparison had been made for SIMILE and BLEU. The plot showed the expected BLEU cost when training ...
3
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1answer
39 views

Shouldn't expected return be calculated for some faraway time in the future $t+n$ instead of current time $t$?

I am learning RL for the first time. It may be naive, but it is a bit odd to grasp this idea that, if the goal of RL is to maximize the expected return, then shouldn't the expected return be ...
2
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2answers
48 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
36 views

What does the notation ${s'\sim T(s,a,\cdot)}$ mean?

I have been seeing notations on Expectations with their respective subscripts such as $E_{s_0 \sim D}[V^\pi (s_0)] = \Sigma_{t=0}^\infty[\gamma^t\phi(s_t)]$. This equation is taken from https://ai....
1
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1answer
99 views

How does the policy gradient's derivative work?

I am trying to understand the policy gradient method using a PyTorch implementation and this tutorial. My first question is about the end result of this gradient derivation, \begin{aligned} \nabla \...
2
votes
1answer
41 views

Why is the expectation calculated over finite number of points drawn from a probability distribution?

This is from the book Pattern Recognition by Bishop. Why is expectation here a simple average? Why is $f(x)$ not being multiplied by $p(x)$?
0
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1answer
59 views

Problem with Proposition 1 of Google Deepmind's 'Weight uncertainty in Neural Networks'

I'm going through the paper Weight Uncertainty in Neural Networks by Google Deepmind. In the final line of the proof of proposition 1, the integral and the derivative are swapped. Then the derivative ...
1
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0answers
39 views

Where does the expectation term in the derivative of the soft-max policy come from?

At slide 17 of the David Silver's series, the soft-max policy is defined as follows $$ \pi_\theta(s, a) \propto e^{\phi(s, a)^T \theta} $$ that is, the probability of an action $a$ (in state $s$) is ...