Questions tagged [expectation]
For questions related to the mathematical concept of "expectation" or "expected value".
21
questions
1
vote
1answer
37 views
What are the iid random variables for a dataset in the GAN framework?
I am trying to understand why mean is used for expectation in training Generative Adversarial Networks.
The answer tells that it is due to the law of large numbers which is based on the assumption ...
3
votes
1answer
165 views
How is the state-value function expressed as a product of sums?
The state-value function for a given policy $\pi$ is given by
$$\begin{align}
V^{\pi}(s) &=E_{\pi}\left\{r_{t+1}+\gamma r_{t+2}+\gamma^{2} r_{t+3}+\cdots \mid s_{t}=s\right\} \\
&=E_{\pi}\...
0
votes
1answer
118 views
What is the meaning of these equations in Noise2Noise paper?
I am trying to understand what is meant by following equations in the Noise2Noise paper by Nvidia.
What is meant by the equation in this image? What is $\mathbb{E}_y\{y\}$? And how should I try to ...
3
votes
1answer
111 views
What is wrong with equation 7.3 in Sutton & Barto's book?
Equation 7.3 of Sutton Barto book:
$$\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}...
5
votes
1answer
70 views
Why is the mean used to compute the expectation in the GAN loss?
From Goodfellow et al. (2014), we have the adversarial loss:
$$ \min_G \, \max_D V (D, G) = \mathbb{E}_{x∼p_{data}(x)} \, [\log \, D(x)] \\
\quad\quad\quad\quad\quad\quad\quad + \, \mathbb{E}_{z∼p_z(...
2
votes
2answers
396 views
How is per-decision importance sampling derived in Sutton & Barto's book?
In per-decison importance sampling given in Sutton & Barto's book:
Eq 5.12 $\rho_{t:T-1}R_{t+k} = \frac{\pi(A_{t}|S_{t})}{b(A_{t}|S_{t})}\frac{\pi(A_{t+1}|S_{t+1})}{b(A_{t+1}|S_{t+1})}\frac{\pi(...
0
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0answers
94 views
Problem in understanding equation given for convergence of TD(n) algorithm
Given equation 7.3 of Sutton and Barto's book for convergence of TD(n):
$\max_s|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)| \leqslant \gamma^n \max_s|V_{t+n-1}(s) - v_\pi(s)|$
$\textbf{PROBLEM ...
3
votes
1answer
90 views
How does $\mathbb{E}$ suddenly change to $\mathbb{E}_{\pi'}$ in this equation?
In Sutton-Barto's book on page 63 (81 of the pdf):
$$\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]$$
How does $...
4
votes
1answer
460 views
If the current state is $S_t$ and the actions are chosen according to $\pi$, what is the expectation of $R_{t+1}$ in terms of $\pi$ and $p$?
I'm trying to solve exercise 3.11 from the book Sutton and Barto's book (2nd edition)
Exercise 3.11 If the current state is $S_t$ , and actions are selected according to a stochastic policy $\pi$, ...
4
votes
2answers
115 views
Why is $G_{t+1}$ is replaced with $v_*(S_{t+1})$ in the Bellman optimality equation?
In equation 3.17 of Sutton and Barto's book:
$$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
votes
1answer
111 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
votes
1answer
56 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
votes
1answer
42 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
votes
2answers
67 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
40 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
vote
1answer
206 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
47 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)$?
4
votes
2answers
231 views
What is the difference between return and expected return?
At a time step $t$, for a state $S_{t}$, the return is defined as the discounted cumulative reward from that time step $t$.
If an agent is following a policy (which in itself is a probability ...
1
vote
1answer
80 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
vote
0answers
43 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}
$$
that is, the probability of an action $a$ (in state $s$) is ...
4
votes
2answers
805 views
What does the argmax of the expectation of the log likelihood mean?
What does the following equation mean? What does each part of the formula represent or mean?
$$\theta^* = \underset {\theta}{\arg \max} \Bbb E_{x \sim p_{data}} \log {p_{model}(x|\theta) }$$
