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.

27 questions with no upvoted or accepted answers
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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', ...
4
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0answers
67 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 ...
4
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2answers
82 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 ...
4
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1answer
140 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 ...
3
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1answer
107 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 ...
2
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1answer
70 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 ...
2
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0answers
23 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 \...
2
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0answers
25 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?
2
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0answers
113 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 ...
2
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0answers
88 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....
2
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1answer
104 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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0answers
75 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
275 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
52 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
47 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}\...
2
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0answers
43 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
2
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0answers
59 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 ...
1
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1answer
65 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_{\...
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0answers
31 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_{...
1
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0answers
28 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 ...
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0answers
74 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 ...
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0answers
56 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 ...
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0answers
51 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.
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1answer
236 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?
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21 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 ...
0
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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 ...
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0answers
63 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 ...