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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17
votes
3answers
2k 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 $\...
16
votes
3answers
5k 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 ...
11
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5answers
3k views

Is there a rigorous proof that AGI is possible, at least, in theory?

It is often implicitly assumed in computer science that the human mind, or at least some mechanical calculations that humans perform (see the Church-Turing thesis), can be replicated with a Turing ...
11
votes
2answers
1k 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-...
10
votes
1answer
760 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 ...
10
votes
1answer
12k 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 ...
9
votes
2answers
785 views

Why is baseline conditional on state at some timestep unbiased?

In the homework for the Berkeley RL class, problem 1, it asks you to show that the policy gradient is still unbiased if the baseline subtracted is a function of the state at time step $t$. $$ \...
7
votes
1answer
440 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. ...
6
votes
1answer
140 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 ...
6
votes
1answer
171 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. ...
6
votes
0answers
159 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', ...
5
votes
2answers
245 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, ...
5
votes
2answers
193 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-...
5
votes
1answer
472 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?
5
votes
2answers
309 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 ...
5
votes
2answers
505 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 ...
5
votes
1answer
271 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 ...
5
votes
2answers
148 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} \...
5
votes
1answer
152 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 ...
5
votes
1answer
106 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 ...
4
votes
1answer
5k 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?
4
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3answers
1k views

Is there a mathematical proof that shows that certain parameters work “better” than others for a certain task?

The machine learning community often only provides empirical results, but I am also interested in theoretical results and proofs. Specifically, is there a mathematical proof that shows that certain ...
4
votes
2answers
2k 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 ...
4
votes
2answers
99 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 ...
4
votes
1answer
518 views

How does L2 regularization make weights smaller?

I'm learning logistic regression and $L_2$ regularization. The cost function looks like below. $$J(w) = -\displaystyle\sum_{i=1}^{n} (y^{(i)}\log(\phi(z^{(i)})+(1-y^{(i)})\log(1-\phi(z^{(i)})))$$ And ...
4
votes
1answer
171 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 ...
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 ...
4
votes
2answers
83 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
votes
1answer
141 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
votes
2answers
74 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 ...
3
votes
1answer
269 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 ...
3
votes
2answers
79 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 ...
3
votes
1answer
288 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-...
3
votes
1answer
291 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
1answer
124 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 ...
3
votes
1answer
86 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
86 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 ...
3
votes
1answer
113 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
votes
1answer
141 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,...
2
votes
2answers
538 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?
2
votes
1answer
535 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
121 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
1answer
281 views

What is the optimal value function of the scaled version of the reward function?

Consider the reward function $r(s, a)$ with optimal state-action value function $q_*(s, a)$. What would be the optimal state-action value function of $c r(s, a)$, for $c \in \mathbb{R}$? Would it be $...
2
votes
1answer
148 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 ...
2
votes
1answer
298 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 $...
2
votes
1answer
1k 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) ...
2
votes
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 ...
2
votes
1answer
94 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
1answer
34 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
48 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 ...