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.

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6
votes
2answers
206 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} \...
2
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0answers
87 views

Where do the characteristics of self-attention come into play in Linformer's proof that self-attention is low rank?

In Linformer's proof that self-attention is low rank in their paper, I don't see how it doesn't generalize to every matrix. They don't utilize any specifics of self-attention (the entire proof feels ...
2
votes
1answer
903 views

Is there a simple proof of the convergence of TD(0)?

Does anybody know a simple proof of the convergence of the TD(0) value function prediction algorithm?
4
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2answers
122 views

What is the proof that the branch and bound algorithm always finds optimal path in a graph?

I've been studying Branch and Bound's graph algorithm and I hear it always finds the optimal path because it uses previously found solutions to find others However, I haven't been able to find a ...
2
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0answers
40 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 ...
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 ...
0
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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
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1answer
31 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
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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 ...
2
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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 ...
2
votes
1answer
469 views

Why does Q-learning converge to the optimal policy, even if the agent acts sub-optimally?

In Q-learning, during training, it doesn't matter how the agent selects actions. The algorithm always converges to the optimal policy. Why does this happen? What's the intuition?
0
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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_\...
0
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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 ...
3
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1answer
132 views

Understanding proof of lemma 1 (policy improvement bound) of the "Trust Region Policy Optimization" paper

In the Trust Region Policy Optimization paper, in Lemma 1 of Appendix A, I did not quite understand the transition from (21) from (20). In going from (20) to (21), $A^\pi(s_t, a_t)$ is substituted ...
1
vote
1answer
72 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_{\...
0
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0answers
32 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
44 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),\...
2
votes
0answers
67 views

How to Prove This Inequality, Related to Generalization Error (Not Using Rademacher Complexity)?

This is an inequality on page 36 of the Foundations of Machine Learning by Mohri, but the author only states it without proof. $$ \mathbb{P}\left[\left|R(h)-\widehat{R}_{S}(h)\right|>\epsilon\right]...
5
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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
36 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
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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 $\...
3
votes
1answer
80 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
115 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 ...
2
votes
1answer
278 views

How do I prove that $\mathcal{H}$, with $\mathcal{VC}$ dimension $d$, shatters all subsets with size less than $d-1$?

If a certain hypothesis class $\mathcal{H}$ has a $\mathcal{VC}$ dimension $d$ over a domain $X$, how can I prove that $H$ will shatter all subsets of $X$ with size less than $d$, i.e. $\mathcal{H}$ ...
2
votes
2answers
141 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{...
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} ...
13
votes
1answer
16k views

Why is A* optimal if the heuristic function is admissible?

A heuristic is admissible if it never overestimates the true cost to reach the goal node from $n$. If a heuristic is consistent, then the heuristic value of $n$ is never greater than the cost of its ...
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
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, ...
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 $\gamma < 1$...
2
votes
1answer
50 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?
4
votes
2answers
4k views

If an heuristic is not admissible, can it be consistent?

I am solving a problem in which, according to the given values, the heuristic is not admissible. According to my calculation from other similar problems, it should be consistent, as well as keeping in ...
2
votes
0answers
31 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
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0answers
18 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
130 views

Is there a mathematical theory behind why MLP can classify handwritten digits?

I'm trying to really understand how multi-layer perceptrons work. I want to prove mathematically that MLP's can classify handwritten digits. The only thing I really have is that each perceptron can ...
2
votes
1answer
89 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 ...
0
votes
1answer
80 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
27 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 ...
4
votes
1answer
187 views

Does Rice's theorem prove safe AI is undecidable?

According to Wikipedia In computability theory, Rice's theorem states that all non-trivial, semantic properties of programs are undecidable. A semantic property is one about the program's ...
5
votes
2answers
349 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-...
2
votes
1answer
740 views

Why isn't Nilsson's Sequence Score an admissible heuristic function?

I understand what an admissible heuristic is, I just don't know how to tell whether one heuristic is admissible or not. So, in this case, I'd like to know why Nilsson's sequence score heuristic ...
4
votes
1answer
647 views

How does L2 regularization make weights smaller?

I'm learning logistic regression and $L_2$ regularization. The cost function looks like below. $$J(w) = -\displaystyle\sum_{i=1}^{n} (y^{(i)}\log(\phi(z^{(i)})+(1-y^{(i)})\log(1-\phi(z^{(i)})))$$ And ...
3
votes
1answer
204 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 ...
5
votes
2answers
377 views

How are the reward functions $R(s)$, $R(s, a)$ and $R(s, a, s')$ equivalent?

In this video, the lecturer states that $R(s)$, $R(s, a)$ and $R(s, a, s')$ are equivalent representations of the reward function. Intuitively, this is the case, according to the same lecturer, ...
18
votes
3answers
6k views

Where can I find the proof of the universal approximation theorem?

The Wikipedia article for the universal approximation theorem cites a version of the universal approximation theorem for Lebesgue-measurable functions from this conference paper. However, the paper ...
5
votes
1answer
387 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 ...
5
votes
0answers
71 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
101 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
3
votes
1answer
539 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-...
11
votes
5answers
3k views

Is there a rigorous proof that AGI is possible, at least, in theory?

It is often implicitly assumed in computer science that the human mind, or at least some mechanical calculations that humans perform (see the Church-Turing thesis), can be replicated with a Turing ...