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.
99
questions
2
votes
0answers
33 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:
$$
\left[(1-\gamma)\left(I-\gamma P^{\pi}\right)^{-1}\right]_{(s, a),\...
5
votes
0answers
46 views
Proof that there always exists a dominating policy in an MDP
I think that it is common knowledge that for any infinite horizon discounted MDP $(S, A, P, r, \gamma)$, there always exists a dominating policy $\pi$, i.e. a policy $\pi$ such that for all policies $\...
2
votes
1answer
27 views
Why adding a baseline doesn't affect the policy gradient?
On the OpenAI's Spinning Up, they justify the fact that adding a baseline $b(s_t)$ in the policy gradient doesn't change its gradient by saying that this is
an immediate consequence of the EGLP Lemma
...
0
votes
0answers
19 views
How should we interpret “common coarsening” in this proof of the uniqueness of coarsest bisimulation?
On page 4 of this pdf in a theoretical RL course, we have a proof of the uniquness of the coarsest bisimulation.
A bisimulation $\phi$ is a mapping from states $s \in\mathcal{S}$ to abstract states $\...
2
votes
1answer
73 views
How to prove the second form of Bellman's equation?
I'd like to prove this "second form" of Bellman's equation: $v(s) = \mathbb{E}[R_{t + 1} + \gamma v(S_{t+1}) \mid S_{t} = s]$ starting from Bellman's equation: $v(s) = \mathbb{E}[G_{t} \mid ...
2
votes
2answers
124 views
Why is the equation $\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{Var} (\epsilon)$ true?
In the book An Introduction to Statistical Learning, the authors claim (equation 2.3, p. 19, chapter 2)
$$\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{...
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
1answer
49 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}
...
2
votes
1answer
52 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$...
1
vote
0answers
33 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, ...
1
vote
0answers
27 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
votes
0answers
15 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
45 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?
0
votes
1answer
65 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
24 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 ...
1
vote
1answer
66 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_{\...
2
votes
1answer
80 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 ...
5
votes
2answers
271 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-...
3
votes
1answer
164 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 ...
4
votes
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 ...
2
votes
1answer
96 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
2
votes
0answers
27 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 \...
1
vote
1answer
31 views
How do I prove that the MSE is zero when all predictions are equal to the corresponding labels?
In the back-propogation algorithm, the error term is:
$$
E=\frac{1}{2}\sum_k(\hat{y}_k - y_k)^2,
$$
where $\hat{y}_k$ is a vector of outputs from the network, $y_k$ is the vector of correct labels (...
0
votes
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 ...
2
votes
2answers
728 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 ...
5
votes
2answers
322 views
Given two optimal policies, is an affine combination of them also optimal?
If there are two different optimal policies $\pi_1, \pi_2$ in a reinforcement learning task, will the linear combination (or affine combination) of the two policies $\alpha \pi_1 + \beta \pi_2, \alpha ...
2
votes
2answers
765 views
If uniform cost search is used for bidirectional search, is it guaranteed the solution is optimal?
If uniform cost search is used for both the forward and backward search in bidirectional search, is it guaranteed the solution is optimal?
6
votes
1answer
261 views
Why does a negative reward for every step really encourage the agent to reach the goal as quickly as possible?
If we shift the rewards by any constant (which is a type of reward shaping), the optimal state-action value function (and so optimal policy) does not change. The proof of this fact can be found here.
...
0
votes
1answer
121 views
What is the optimal value function of the shifted version of the reward function?
Similarly to this question that I asked some time ago, what is the optimal value function of the shifted (by some constant $c$) version of some reward function? More precisely, let's assume that $r(s, ...
3
votes
2answers
76 views
Why does (not) the distribution of states depend on the policy parameters that induce it?
I came across the following proof of what's commonly referred to as the log-derivative trick in policy-gradient algorithms, and I have a question -
While transitioning from the first line to the ...
2
votes
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?
1
vote
0answers
34 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
$$V(S_t) = V(S_{t-1}) + \alpha \rho_{...
2
votes
2answers
89 views
Can a computer make a proof by induction?
Can a computer solve the following problem, i.e. make a proof by induction? And why?
Prove by induction that $$\sum_{k=1}^nk^3=\left(\frac{n(n+1)}{2}\right)^2, \, \, \, \forall n\in\mathbb N .$$
I'm ...
5
votes
2answers
159 views
Why are the Bellman operators contractions?
In these slides, it is written
\begin{align}
\left\|T^{\pi} V-T^{\pi} U\right\|_{\infty} & \leq \gamma\|V-U\|_{\infty} \tag{9} \label{9} \\
\|T V-T U\|_{\infty} & \leq \gamma\|V-U\|_{\infty} \...
6
votes
0answers
165 views
Is the Bellman equation that uses sampling weighted by the Q values (instead of max) a contraction?
It is proved that the Bellman update is a contraction (1).
Here is the Bellman update that is used for Q-Learning:
$$Q_{t+1}(s, a) = Q_{t}(s, a) + \alpha*(r(s, a, s') + \gamma \max_{a^*} (Q_{t}(s',
...
1
vote
0answers
29 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
54 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 ...
3
votes
1answer
384 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-...
2
votes
0answers
136 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
78 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
290 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 ...
6
votes
1answer
163 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
1answer
52 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
137 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 ...
6
votes
2answers
719 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
36 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
52 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 ...
4
votes
1answer
169 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
259 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. $\mathcal{H}$ ...
1
vote
3answers
156 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 + ...
