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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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),\...
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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 $\...
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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 ...
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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 $\...
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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 ...
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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{...
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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? ...
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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} ...
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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 $\gamma < 1$...
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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, ...
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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 ...
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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?
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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?
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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?
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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 ...
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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_{\...
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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 ...
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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-...
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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 ...
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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 ...
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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
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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 \...
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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 (...
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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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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 ...
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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 ...
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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?
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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. ...
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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, ...
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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 ...
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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?
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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_{...
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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 ...
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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
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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', ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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. $\mathcal{H}$ ...
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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 + ...