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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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4
votes
2answers
101 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
23 views

How to show Sauer's Lemma when the inequalities are strict or they are equalities?

I have the following homework. We proved Sauer's lemma by proving that for every class $H$ of finite VC-dimension $d$, and every subset $A$ of the domain, $$ \left|\mathcal{H}_{A}\right| \leq |\...
6
votes
1answer
53 views

Can two admissable heuristics not dominate each other?

I am working on a project for my artificial intelligence class. I was wondering if I have 2 admissible heuristics, A and B, is it possible that A does not dominate B and B does not dominate A? I am ...
8
votes
1answer
369 views

What are the implications of the “No Free Lunch” theorem for machine learning?

The No Free Lunch (NFL) theorem states (see the paper Coevolutionary Free Lunches by David H. Wolpert and William G. Macready) any two algorithms are equivalent when their performance is averaged ...
2
votes
1answer
40 views

Why does estimation error increase with $|H|$ and decrease with $m$ in PAC learning?

Why does estimation error increase with $|H|$ and decrease with $m$ in PAC learning? I came across this statement in the section 5.2 of the book "understanding machine learning: from theory to ...
2
votes
1answer
50 views

What happens to the optimal value function if the reward is multiplied by a constant?

What happens to the optimal action-value function, $q_*$ if the reward is multiplied by a constant $c$? Is the optimal action-value function also multiplied by such a constant?
2
votes
0answers
42 views

How to show Monte Carlo methods converge to an estimate which minimizes mean squared error?

In chapter six of Sutton and Barto (p.128), they claim Monte Carlo methods converge to an estimate minimizing the mean squared error. How can this be shown formally? Bump
3
votes
1answer
100 views

How to show temporal difference methods converge to MLE?

In chapter 6 of Sutton and Barto (p. 128), they claim temporal difference converges to the maximum likelihood estimate (MLE). How can this be shown formally?
2
votes
1answer
43 views

Is unsupervised disentanglement really impossible?

In Locatello et al's Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations he claims to prove unsupervised disentanglement is impossible. His entire claim is ...
1
vote
1answer
46 views

Is there a rigorous proof for finding Hopfield minima?

I am looking for a rigorous mathematical proof for finding the several local minima of the Hopfield networks. I am searching for something rigorous, a demonstration, not just let the network keep ...
4
votes
1answer
123 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 ...
2
votes
1answer
66 views

Understanding the proof that A* search is optimal

I don't understand the proof that $A^*$ is optimal. The proof is by contradiction: Assume $A^*$ returns $p$ but there exists a $p'$ that is cheaper. When $p$ is chosen from the frontier, assume $...
3
votes
1answer
89 views

How is G(z) related to x in GAN proof?

In the proofs for the original GAN paper, it is written: $$∫_x p_{data}(x) \log D(x)dx+∫_zp(z)\log(1−D(G(z)))dz =∫_xp_{data}(x)\log D(x)+p_G(x) \log(1−D(x))dx$$ I've seen some explanations asserting ...
0
votes
0answers
34 views

Proof of Correctness of Monte Carlo Tree Search

I'm trying to make the proof of correctness of Monte Carlo Tree Search. Any help would be really appreciated.
3
votes
1answer
26 views

Proof Branch and Bound 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, but I haven't been able to find a proof on ...
2
votes
1answer
81 views

Is the minimum and maximum of a set of admissible and consistent heuristics also consistent and admissible?

Let's suppose I have a set of heuristics $H$ = {$h_1, h_2, ..., h_N$}. If all heuristics in $H$ are admissible, does that mean that a heuristic that takes the $\min(H)$ (or $\max(H)$ for that matter) ...
10
votes
2answers
257 views

Why doesn't Q-learning converge when using function approximation?

The tabular Q-learning algorithm is guaranteed to find the optimal $Q$ function, $Q^*$, provided the following conditions regarding the learning rate are satisfied $\sum_{t} \alpha_t(s, a) = \infty$ $...
3
votes
1answer
133 views

How do I find whether this heuristic is or not admissible and consistent?

I was given the following problem to solve. Given a circular trail divided by $n> 2$ segments labeled $0 \dots n-1$. In the beginning, an agent is at the start of segment number $0$ (the edge ...
3
votes
2answers
55 views

Why is the max a non-expansive operator?

In certain reinforcement learning (RL) proofs, the operators involved are assumed to be non-expansive. For example, on page 6 of the paper Generalized Markov Decision Processes: Dynamic-programming ...
2
votes
1answer
147 views

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

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 heuristic isn't admissible.
4
votes
2answers
68 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, ...
6
votes
1answer
557 views

How do we prove the n-step return error reduction property?

In section 7.1 (about the n-step bootstrapping) of the book Reinforcement Learning: An Introduction (2nd edition), by Andrew Barto and Richard S. Sutton, the authors write about what they call the "n-...
1
vote
1answer
82 views

Understanding lemma 2 of the “Trust Region Policy Optimization” paper

In the Trust Region Policy Optimization paper, in Lemma 2 of Appendix A, I did not quite understand deriving inequality (31) from (30), which is: $$\bar{A}(s) = P(a \neq \tilde{a} | s) \mathbb{E}_{(a,...
4
votes
1answer
2k views

How do I show that uniform-cost search is a special case of A*?

How do I show that uniform-cost search is a special case of A*? How do I prove this?
1
vote
0answers
70 views

Why does Q-learning converges to optimal policy even if I am acting suboptimally?

In Q-learning, during training, it doesn’t matter how I select actions. The algorithm always converges to optimal optimal policy. Why does this happen?
2
votes
0answers
39 views

Understanding the proof of theorem 2.1 from the paper “Efficient reductions for imitation learning”

I am trying to understand the proof of theorem 2.1 from this paper: Ross, Stéphane, and Drew Bagnell. "Efficient reductions for imitation learning." Proceedings of the thirteenth international ...
5
votes
1answer
58 views

Is there a limit of minimum error for a particular training dataset in artificial Neural Network?

In error-based learning using gradient descent, if I give you a training dataset, then can you find the minimum error after training? And the minimum error should be true for all architectures of a ...
6
votes
2answers
314 views

How can a neural network approximate all functions when the weights are not allowed to grow exponentially?

It has been proven in the paper "Approximation by Superpositions of a Sigmoidal Function" (by Cybenko, in 1989) that neural networks are universal function approximators. I have a related question. ...
1
vote
1answer
229 views

Why is exact inference in a Bayesian network both NP-hard and P-hard?

I should show that exact inference in a Bayesian network (BN) is NP-hard and P-hard by using a 3-SAT problem. So, I did formulate a 3-SAT problem by defining 3-CNF: $$(x_1 \lor x_2) \land (\neg x_3 \...
7
votes
1answer
6k 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 ...
-1
votes
2answers
88 views

When is a measurable function a Bayesian decision function?

When is a measurable function a Bayesian decision function? How do I prove this? Can you give an example with standard or weighted binary classification?
11
votes
9answers
2k views

Proof that Artificial General Intelligence is possible

It is assumed in computer science that the human mind can be replicated with a Turing machine, therefore Artificial General Intelligence (AGI) is possible. To assume otherwise is to believe in ...
2
votes
1answer
777 views

Understanding why the expectation is over the new policy $\pi'$ in the proof of the Policy Improvement Theorem

In reinforcement learning, policy improvement is a part of an algorithm called policy iteration, which attempts to find approximate solutions to the Bellman optimality equations. Pages 84 and 85 in ...
-2
votes
3answers
196 views

Is there a proof that states that an AI can become smarter than its creator?

Is there a proof that states the possibility or impossibility of an AI system to acquire more sophisticated capabilities (in terms of generic cleverness) than its own creator?