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.
63
questions
1
vote
0answers
16 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
35 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 ...
1
vote
0answers
43 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-...
1
vote
0answers
54 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
28 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
114 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 ...
1
vote
0answers
20 views
Why is it hard to prove the convergence of the deep Q-learning algorithm?
Why is it hard to prove the convergence of the DQN algorithm? We know that the tabular Q-learning algorithm converges to the optimal Q-values, and with a linear approximator convergence is proved.
...
1
vote
1answer
39 views
Why is probability that at least one hypothesis out of $k$ being consistent with $m$ training examples $k(1- \epsilon)^m$?
My question is actually related to the addition of probabilities. I am reading on computational learning theory from Tom Mitchell's machine learning book.
In chapter 7, when proving the upper bound ...
1
vote
1answer
47 views
Monte Carlo epsilon-greedy Policy Iteration: monotonic improvement for all cases or for the expected value?
I was going through university slides and this particular slide is trying to prove that in a Monte Carlo Policy Iteration algorithm using an epsilon-greedy policy, the state Values (V-Values) are ...
3
votes
2answers
79 views
What is the proof that policy evaluation converges to the optimal solution?
Although I know how the algorithm of iterative policy evaluation using dynamic programming works, I am having a hard time realizing how it actually converges.
It appeals to intuition that, with each ...
2
votes
1answer
22 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 \...
2
votes
1answer
37 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
votes
0answers
61 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
votes
1answer
79 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 \...
1
vote
3answers
87 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
$$f(x_1, x_2, x_3, x_4) = (x_1 + 10x_2)^2 + 5x_2^3 + (x_2 + 2x_3)^4 + ...
2
votes
0answers
64 views
Is an oracle that answers only with a “yes” or “no” dangerous?
I was thinking about the risks of Oracle AI and it doesn't seem as safe to me as Bostrom et al. suggest. From my point of view, even an AGI that only answers questions could have a catastrophic impact....
3
votes
1answer
89 views
Is the summation of consistent heuristic functions also consistent?
Imagine that we have a set of heuristic functions $\{h_i\}_{i=1}^N$, where each $h_i$ is both admissible and consistent (monotonic). Is $\sum_{i=1}^N h_i$ still consistent or not?
Is there any proof ...
1
vote
0answers
21 views
Does SARSA(0) converge to the optimal policy in expectation if the Robbins-Monro conditions are removed?
The conditions of convergence of SARSA(0) to the optimal policy are :
The Robbins-Monro conditions above hold for $Ī±_t$.
Every state-action pair is visited infinitely often
The policy is greedy with ...
1
vote
2answers
36 views
What are the conditions for the convergence of SARSA to the optimal value function?
Is it correct that for SARSA to converge to the optimal value function (and policy)
The learning rate parameter $\alpha$ must satisfy the conditions:
$$\sum \alpha_{n^k(s,a)} =\infty \quad \text{and}...
1
vote
1answer
62 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?
$$\sum \alpha_t =\infty \quad \text{...
1
vote
1answer
160 views
Is there a simple proof of the convergence of TD(0)?
Does anybody know a simple proof of the convergence of the TD(0) value function prediction algorithm?
2
votes
1answer
84 views
Is there a mathematical theory behind why MLP can classify handwritten digits?
I'm trying to really understand how multi-layer perceptrons work. I want to prove mathematically that MLP's can classify handwritten digits. The only thing I really have is that each perceptron can ...
2
votes
1answer
67 views
Why does KL divergence not satisfy the triangle inequality?
$D_{KL}=\sum_i p(x_i)log(p(x_i)/q(x_i)$
Also can't you make it satisfy the triangle inequality by taking the absolute value of the information at every point?
2
votes
0answers
28 views
How can I show that the VC dimension of the set of all closed balls in $\mathbb{R}^n$ is at most $n+3$?
How can I show that the VC dimension of the set of all closed balls in $\mathbb{R}^n$ is at most $n+3$?
For this problem, I only try the case $n=2$ for 1. When $n=2$, consider 4 points $A,B,C,D$ and ...
2
votes
0answers
61 views
A problem about the relation between 1-oracle and 2-oracle PAC model
This problem is about two-oracle variant of the PAC model. Assume that positive and negative examples are now drawn from two separate distributions $\mathcal{D}_{+}$ and $\mathcal{D}_{-} .$ For an ...
2
votes
0answers
33 views
How can we prove this inequality, related to the generalization error, without using the Rademacher complexity?
This is an inequality on page 36 of the book Foundations of Machine Learning, but the author only states it without proof.
$$
\mathbb{P}\left[\left|R(h)-\widehat{R}_{S}(h)\right|>\epsilon\right] \...
2
votes
0answers
26 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
votes
1answer
54 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
votes
1answer
72 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
votes
2answers
812 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
votes
1answer
71 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
votes
1answer
575 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
votes
2answers
92 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
182 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
208 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
votes
1answer
62 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
77 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 ...
11
votes
1answer
2k 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
109 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 $...
4
votes
1answer
136 views
How is G(z) related to x in GAN proof?
In the proofs for the original GAN paper, it is written:
$$ā«_x p_{data}(x) \log D(x)dx+ā«_zp(z)\log(1āD(G(z)))dz
=ā«_xp_{data}(x)\log D(x)+p_G(x) \log(1āD(x))dx$$
I've seen some explanations asserting ...
1
vote
0answers
45 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.
4
votes
2answers
45 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
votes
1answer
339 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) ...
14
votes
3answers
834 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
votes
1answer
202 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
62 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
286 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
99 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, ...
8
votes
2answers
843 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-...
