Questions tagged [proofs]
For questions that ask about or call for proofs for specific assertions, whether they be proofs of theorems or corollaries, proofs of concept through working implementation, counter proofs, or counter examples.
107
questions
2
votes
0answers
33 views
Why does TD (0) converge to the MLE solution of the Markov model?
Why does TD (0) converge to the MLE solution of the Markov model?
Let's take the Example 6.4 in Sutton and Barto's book as an example.
Example 6.4: You are the Predictor Place yourself now in the ...
0
votes
0answers
17 views
proof of convergence for the random forest algorithm
I am looking for the proof of convergence of the random forest algorithm. A cursory google search shows many, but I do not understand which version (original?) of the algorithm this is.
Can you kindly ...
0
votes
1answer
29 views
How to show $\rho > 0$ when $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$, where $W^*$ the vector that separates the data?
In the book Learning from Data written (by Abu Mostafa), we have the following exercise:
Let $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$ where $W^*$ is the vector that separates the data. Show ...
1
vote
0answers
57 views
What is the derivative of equation 1 in the paper "Conservative Q-Learning for Offline Reinforcement Learning"?
I am looking at the paper Conservative Q-Learning for Offline Reinforcement Learning, but I'm not sure how they proved theorem 3.1.
Here is a screenshot of theorem 3.1.
In the proof of theorem 3.1
...
0
votes
1answer
31 views
Policies for which the policy improvement theorem holds
According to Reinforcement Learning (2nd Edition) by Sutton and Barto, the policy improvement theorem states that for any pair of deterministic policies $\pi'$ and $\pi$, if $q_\pi(s,\pi'(s)) \geq v_\...
2
votes
1answer
31 views
How to prove importance sampling ratio is uncorrelated with action-value (or state-value) estimate?
In Sutton & Barto (2nd edition), the following is mentioned on page 150 (p. 172 of the pdf), section 7.4:
the importance sampling ratio has expected value one (Section 5.9) and is uncorrelated ...
0
votes
0answers
29 views
Non-locally Electrically Programmable Logic Gates - Technological Advances Progress
Preface: I’d like to clarify that I understand what a relay is and that a PLC uses a fairly conventional microprocessor that only digitally establishes logical logic gate configuration as a digitally ...
0
votes
0answers
31 views
Why is the proof of convergence in the GAN paper not applicable practically?
This question is about generative adversarial networks and restricted to the research paper titled Generative Adversarial Nets.
If I select a particular architecture of MLP as a generator and trained ...
2
votes
0answers
43 views
How to prove Lemma 1.6 in the book "Reinforcement Learning: Theory and Algorithms"
I am trying to prove the following lemma from Reinforcement Learning: Theory and Algorithms on page 8.
Lemma 1.6. We have that:
$$
\left[(1-\gamma)\left(I-\gamma P^{\pi}\right)^{-1}\right]_{(s, a),\...
5
votes
0answers
52 views
Proof that there always exists a dominating policy in an MDP
I think that it is common knowledge that for any infinite horizon discounted MDP $(S, A, P, r, \gamma)$, there always exists a dominating policy $\pi$, i.e. a policy $\pi$ such that for all policies $\...
2
votes
1answer
32 views
Why adding a baseline doesn't affect the policy gradient?
On the OpenAI's Spinning Up, they justify the fact that adding a baseline $b(s_t)$ in the policy gradient doesn't change its gradient by saying that this is
an immediate consequence of the EGLP Lemma
...
0
votes
0answers
19 views
How should we interpret "common coarsening" in this proof of the uniqueness of coarsest bisimulation?
On page 4 of this pdf in a theoretical RL course, we have a proof of the uniquness of the coarsest bisimulation.
A bisimulation $\phi$ is a mapping from states $s \in\mathcal{S}$ to abstract states $\...
2
votes
1answer
76 views
How to prove the second form of Bellman's equation?
I'd like to prove this "second form" of Bellman's equation: $v(s) = \mathbb{E}[R_{t + 1} + \gamma v(S_{t+1}) \mid S_{t} = s]$ starting from Bellman's equation: $v(s) = \mathbb{E}[G_{t} \mid ...
2
votes
2answers
137 views
Why is the equation $\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{Var} (\epsilon)$ true?
In the book An Introduction to Statistical Learning, the authors claim (equation 2.3, p. 19, chapter 2)
$$\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{...
0
votes
0answers
28 views
Given an admissible/consistent heuristic $h(n)$, would the tree and graph search versions of A* return the optimal path with $g(n)=2h(n)$?
I know that, if the heuristic function is admissible and consistent, then the A* algorithm always returns an optimal solution. But for the following situation, does A* search return optimal solution?
...
1
vote
1answer
51 views
When showing that the policy improvement theorem applies to MC control, why is $q_{\pi_{k}}\left(s, \pi_{k}(s)\right) \geq v_{\pi_{k}}(s)$ true?
When discussing why the policy improvement theorem is true, when we do Monte Carlo control by updating greedily, it says on page 98 of Sutton and Barto's book (2nd edition) that:
$$
\begin{aligned}
...
2
votes
1answer
55 views
Is existence and uniqueness of state-value function at $\gamma < 1$ theoretical?
Consider the following statement from 4.1 Policy Evaluation of the first edition of Sutton and Barto's book.
The existence and uniqueness of $V^{\pi}$ are guaranteed as long as
either $\gamma < 1$...
1
vote
0answers
35 views
Why is the number of examined nodes $ O(b^{3d/4})$ in $\alpha$-$\beta$ pruning?
I'm taking a course 'Introduction to AI' and, in one of the tutorials, it was written that when pruning the game tree using $\alpha$-$\beta$ boundaries, the number of nodes that will be developed, ...
1
vote
0answers
29 views
How to prove that a regularisation method simplified a neural network?
There are a few ways to regularise a neural network, for example dropout or the L1. Now, both these methods, and possibly most other regularisation methods, tend to remove from, or simplify the neural ...
0
votes
0answers
16 views
Why is that the heuristic is not admissible, then it is not consistent [duplicate]
I don't understand why if the heuristic is not admissible then it is not consistent?
2
votes
1answer
49 views
Is $\min(h_1(s),\ h_2(s))$ consistent?
If $h_1(s)$ is a consistent heuristic and $h_2(s)$ is a admissible heuristic, is $\min(h_1(s),\ h_2(s))$ consistent?
0
votes
1answer
76 views
Are hill climbing variations always optimal and complete?
Are hill climbing variations (like steepest ascent hill climbing, stochastic hill climbing, random restart hill climbing, local beam search) always optimal and complete?
0
votes
0answers
26 views
How is the discounted maximum entropy objective obtained for soft-q-learning and SAC
In the soft q-learning paper, they provide an expression for the maximum entropy objective that takes discounting into account.
My main question is: can someone explain how they incorporated ...
1
vote
1answer
70 views
Can the law of iterated expectation be used on the inner expectation of the DQN cost function described in the DQN paper
Is the expression for the DQN cost function, Equation (2) of the DQN paper
$$\begin{align}L_1 &= E_{\mu,\pi}\left[\left(y_i - q(s,a;\theta)\right)^2\right]\\
&=E_{\mu,\pi}\left[\left(E_{\...
2
votes
1answer
85 views
If $h_1(n)$ is admissible, why does A* tree search with $h_2(n) = 3h_1(n)$ return a path that is at most thrice as long as the optimal path?
Consider a heuristic function $h_2(n) = 3h_1(n)$. Where $h_1(n)$ is admissible.
Why are the following statements true?
$A^*$ tree search with $h_2(n)$ will return a path that is at most thrice as ...
5
votes
2answers
316 views
Why is "-0.5 * torch.sum(1 + sigma - mu.pow(2) - sigma.exp())" in Pytorch equivalent to the KL?
I found this code for the loss function of a VAE:
-0.5 * torch.sum(1 + sigma - mu.pow(2) - sigma.exp())
From this link: https://debuggercafe.com/getting-started-...
3
votes
1answer
172 views
Is my proof of equation 0.6 in the book "Reinforcement Learning: Theory and Algorithms" correct?
In Sham Kakade's Reinforcement Learning: Theory and Algorithms, this equation (page 17) is used preceding the proof of performance difference lemma.
I am attempting to prove equation 0.6. Here is my ...
4
votes
0answers
68 views
When do two identical neural networks have uncorrelated errors?
In Chapter 9, section 9.1.6, Raul Rojas describes how committees of networks can reduce the prediction error by training N identical neural networks and averaging the results.
If $f_i$ are the ...
2
votes
1answer
99 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
2
votes
0answers
28 views
Does there necessarily exist "dominated actions" in a MDP?
In a Markov Decision Process, is it possible that there exists no "dominated action"?
I define a dominated action the following way:
we say that $(s,a)$ is a dominated action, if $\forall \...
1
vote
1answer
32 views
How do I prove that the MSE is zero when all predictions are equal to the corresponding labels?
In the back-propogation algorithm, the error term is:
$$
E=\frac{1}{2}\sum_k(\hat{y}_k - y_k)^2,
$$
where $\hat{y}_k$ is a vector of outputs from the network, $y_k$ is the vector of correct labels (...
0
votes
0answers
17 views
Is the PR AUC invariant under label flip?
The ROC-AUC curve is invariant under a flip of the labels. I don't know if it's a famous result, so I will give the proof below. My question is if the PR-AUC curve also has this property. I have not ...
2
votes
2answers
1k views
Is there a proof to explain why XOR cannot be linearly separable?
Can someone explain to me with a proof or example why you can't linearly separate XOR (and therefore need a neural network, the context I'm looking at it in)?
I understand why it's not linearly ...
5
votes
2answers
332 views
Given two optimal policies, is an affine combination of them also optimal?
If there are two different optimal policies $\pi_1, \pi_2$ in a reinforcement learning task, will the linear combination (or affine combination) of the two policies $\alpha \pi_1 + \beta \pi_2, \alpha ...
2
votes
2answers
1k views
If uniform cost search is used for bidirectional search, is it guaranteed the solution is optimal?
If uniform cost search is used for both the forward and backward search in bidirectional search, is it guaranteed the solution is optimal?
6
votes
1answer
363 views
Why does a negative reward for every step really encourage the agent to reach the goal as quickly as possible?
If we shift the rewards by any constant (which is a type of reward shaping), the optimal state-action value function (and so optimal policy) does not change. The proof of this fact can be found here.
...
0
votes
1answer
140 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
votes
2answers
77 views
Why does (not) the distribution of states depend on the policy parameters that induce it?
I came across the following proof of what's commonly referred to as the log-derivative trick in policy-gradient algorithms, and I have a question -
While transitioning from the first line to the ...
2
votes
0answers
26 views
Do we assume the policy to be deterministic when proving the optimality?
In reinforcement learning, when we talk about the principle of optimality, do we assume the policy to be deterministic?
1
vote
0answers
36 views
How can I derive n-step off-policy temporal difference formula?
I was reading the book "Reinforcement Learning: An Introduction" by Sutton and Barto. In section 7.3, they write the formula for n-step off-policy TD as
$$V(S_t) = V(S_{t-1}) + \alpha \rho_{...
2
votes
2answers
105 views
Can a computer make a proof by induction?
Can a computer solve the following problem, i.e. make a proof by induction? And why?
Prove by induction that $$\sum_{k=1}^nk^3=\left(\frac{n(n+1)}{2}\right)^2, \, \, \, \forall n\in\mathbb N .$$
I'm ...
5
votes
2answers
173 views
Why are the Bellman operators contractions?
In these slides, it is written
\begin{align}
\left\|T^{\pi} V-T^{\pi} U\right\|_{\infty} & \leq \gamma\|V-U\|_{\infty} \tag{9} \label{9} \\
\|T V-T U\|_{\infty} & \leq \gamma\|V-U\|_{\infty} \...
6
votes
0answers
170 views
Is the Bellman equation that uses sampling weighted by the Q values (instead of max) a contraction?
It is proved that the Bellman update is a contraction (1).
Here is the Bellman update that is used for Q-Learning:
$$Q_{t+1}(s, a) = Q_{t}(s, a) + \alpha*(r(s, a, s') + \gamma \max_{a^*} (Q_{t}(s',
...
1
vote
0answers
30 views
What is the proof that the variance of the gradient estimate in Actor-Critic is smaller than in REINFORCE?
The intuition provided when introducing actor-critic algorithms is that the variance of its gradient estimates is smaller than in REINFORCE as, e.g., discussed here. This intuition makes sense for the ...
1
vote
1answer
58 views
When to use AND and when to use Implies in first-order logic?
I am trying to learn the theory behind first-order logic (FOL) and do some practice runs of converting statements into the form of FOL.
One issue I keep running into is hesitating on whether to use an ...
3
votes
1answer
490 views
Proof of Maximization Bias in Q-learning?
In the textbook "Reinforcement Learning: An Introduction" by Richard Sutton and Andrew Barto, the concept of Maximization Bias is introduced in section 6.7, and how Q-learning "over-estimates" action-...
2
votes
0answers
152 views
What is the proof that "reward-to-go" reduces variance of policy gradient?
I am following the OpenAI's spinning up tutorial Part 3: Intro to Policy Optimization. It is mentioned there that the reward-to-go reduces the variance of the policy gradient. While I understand the ...
1
vote
0answers
78 views
How do you prove that minimax algorithm outputs a subgame-perfect Nash equilibrium?
At every node, MAX would always move to maximise the minimum payoff while MIN choose to minimise the maximum payoff, hence there is nash equilibrium.
By using backwards induction, at every node, MAX ...
5
votes
1answer
358 views
How do I convert an MDP with the reward function in the form $R(s,a,s')$ to and an MDP with a reward function in the form $R(s,a)$?
The AIMA book has an exercise about showing that an MDP with rewards of the form $r(s, a, s')$ can be converted to an MDP with rewards $r(s, a)$, and to an MDP with rewards $r(s)$ with equivalent ...
6
votes
1answer
169 views
Can deep learning be used to help mathematical research?
I am currently learning about deep learning and artificial intelligence and exploring his possibilities, and, as a mathematician at heart, I am inquisitive about how it can be used to solve problems ...
