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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2
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1answer
21 views

Equivalence between expected parameter increments in “Off-Policy Temporal-Difference Learning with Function Approximation”

I am having a hard time understanding the proof of theorem 1 presented in the "Off-Policy Temporal-Difference Learning with Function Approximation" paper. Let $\Delta \theta$ and $\Delta \bar{\theta}...
2
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1answer
36 views

Is the derivative of the loss wrt a single scalar parameter proportional to the loss?

I am wondering about the correlation between the loss and the derivative of the loss wrt a single scalar parameter, with the same sample. That means: considering a machine learning model with ...
2
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0answers
50 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 ...
2
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1answer
62 views

How to prove $\mathcal H$ with VC dimension $d$ shatter all subsets with size less than $d-1$?

I was wondering that if a certain hypothesis class $H$ has a VC dimension $d$ over domain $X$ how to prove that $H$ will shatter all subsets of $X$ with size less than $d$ i.e $H$ will shatter $A \...
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3answers
81 views

How to prove that gradient descent doesn't necessarily find the global optimum?

How can I prove that gradient descent doesn't necessarily find the global optimum? For example, consider the following function $$f(x_1, x_2, x_3, x_4) = (x_1 + 10x_2)^2 + 5x_2^3 + (x_2 + 2x_3)^4 + ...
2
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0answers
59 views

Is an oracle that answers only with a “yes” or “no” dangerous?

I was thinking about the risks of Oracle AI and it doesn't seem as safe to me as Bostrom et al. suggest. From my point of view, even an AGI that only answers questions could have a catastrophic impact....
3
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1answer
56 views

Is the summation of consistent heuristic functions also consistent?

Imagine that we have a set of heuristic functions $\{h_i\}_{i=1}^N$, where each $h_i$ is both admissible and consistent (monotonic). Is $\sum_{i=1}^N h_i$ still consistent or not? Is there any proof ...
1
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0answers
18 views

Does SARSA(0) converge to the optimal policy in expectation if the Robbins-Monro conditions are removed?

The conditions of convergence of SARSA(0) to the optimal policy are : The Robbins-Monro conditions above hold for $α_t$. Every state-action pair is visited infinitely often The policy is greedy with ...
1
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2answers
31 views

What are the conditions for the convergence of SARSA to the optimal value function?

Is it correct that for SARSA to converge to the optimal value function (and policy) The learning rate parameter $\alpha$ must satisfy the conditions: $$\sum \alpha_{n^k(s,a)} =\infty \quad \text{and}...
1
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1answer
45 views

Does TD(0) prediction require Robbins-Monro conditions to converge to the value function?

Does the learning rate parameter $\alpha$ require the Robbins-Monro conditions below for the TD(0) algorithm to converge to the true value function of a policy? $$\sum \alpha_t =\infty \quad \text{...
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1answer
140 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?
2
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1answer
79 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
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1answer
55 views

Why does KL divergence not satisfy the triangle inequality?

$D_{KL}=\sum_i p(x_i)log(p(x_i)/q(x_i)$ Also can't you make it satisfy the triangle inequality by taking the absolute value of the information at every point?
2
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0answers
24 views

How can I show that the VC dimension of the set of all closed balls in $\mathbb{R}^n$ is at most $n+3$?

How can I show that the VC dimension of the set of all closed balls in $\mathbb{R}^n$ is at most $n+3$? For this problem, I only try the case $n=2$ for 1. When $n=2$, consider 4 points $A,B,C,D$ and ...
2
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0answers
44 views

A problem about the relation between 1-oracle and 2-oracle PAC model

This problem is about two-oracle variant of the PAC model. Assume that positive and negative examples are now drawn from two separate distributions $\mathcal{D}_{+}$ and $\mathcal{D}_{-} .$ For an ...
2
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0answers
31 views

How can we prove this inequality, related to the generalization error, without using the Rademacher complexity?

This is an inequality on page 36 of the book Foundations of Machine Learning, but the author only states it without proof. $$ \mathbb{P}\left[\left|R(h)-\widehat{R}_{S}(h)\right|>\epsilon\right] \...
2
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0answers
24 views

Convert a PAC-learning algorithm into another one which requires no knowledge of the parameter

This is part of the exercise 2.13 in the book Foundations of Machine Learning (page 28). You can refer to chapter 2 for the notations. Consider a family of concept classes $\left\{\mathcal{C}_{s}\...
3
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1answer
51 views

Why is the stationary distribution independent of the initial state in the proof of the policy gradient theorem?

I was going through the proof of the policy gradient theorem here: https://lilianweng.github.io/lil-log/2018/04/08/policy-gradient-algorithms.html#svpg In the section "Proof of Policy Gradient ...
3
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1answer
62 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 ...
4
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2answers
447 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 ...
6
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1answer
64 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 ...
10
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1answer
497 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 ...
4
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2answers
89 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
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1answer
101 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
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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
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1answer
142 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?
3
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1answer
59 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 ...
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1answer
67 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 ...
8
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1answer
946 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
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1answer
87 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
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1answer
121 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 ...
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0answers
44 views

Proof of Correctness of Monte Carlo Tree Search

I'm trying to write the proof of correctness of Monte Carlo Tree Search. Any help would be really appreciated.
3
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1answer
28 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
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1answer
237 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) ...
11
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2answers
527 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 (the Robbins-Monro conditions) regarding the learning rate are satisfied $\...
3
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1answer
185 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
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2answers
60 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
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1answer
219 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
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2answers
78 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
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1answer
714 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-...
2
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1answer
108 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
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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?
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0answers
87 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
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0answers
42 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
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1answer
68 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 ...
7
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1answer
351 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. ...
2
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1answer
279 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 \...
8
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1answer
8k 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 ...
11
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8answers
3k 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
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1answer
1k 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 ...