# 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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### Why does TD (0) converge to the MLE solution of the Markov model?

Why does TD (0) converge to the MLE solution of the Markov model? Let's take the Example 6.4 in Sutton and Barto's book as an example. Example 6.4: You are the Predictor Place yourself now in the ...
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### proof of convergence for the random forest algorithm

I am looking for the proof of convergence of the random forest algorithm. A cursory google search shows many, but I do not understand which version (original?) of the algorithm this is. Can you kindly ...
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### How to show $\rho > 0$ when $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$, where $W^*$ the vector that separates the data?

In the book Learning from Data written (by Abu Mostafa), we have the following exercise: Let $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$ where $W^*$ is the vector that separates the data. Show ...
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### What is the derivative of equation 1 in the paper "Conservative Q-Learning for Offline Reinforcement Learning"?

I am looking at the paper Conservative Q-Learning for Offline Reinforcement Learning, but I'm not sure how they proved theorem 3.1. Here is a screenshot of theorem 3.1. In the proof of theorem 3.1 ...
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### 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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### 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 ...
1k 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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### 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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### 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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### 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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### 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 ...
490 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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### 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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### 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 ...
### 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 ...