# Questions tagged [value-functions]

For questions related to the concept of value (or performance, or quality, or utility) function (as defined in reinforcement learning and other AI sub-fields). An example of this type of functions is the Q function (used e.g. in the Q-learning algorithm), also known as the state-action value function, given that $Q: S \times A \rightarrow \mathbb{R}$, where $S$ and $A$ are respectively the set of states and actions of the environment.

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### Proof that the Policy Iteration Converges?

Let $\mathcal{X}=:\{x_1, x_2, x_3,...,x_n\}$ be the state space. Let $\mathcal{U}:=\{u_1, u_2, u_3,...,u_m\}$ be the set of actions. Let $A^{u_1}, A^{u_2}, A^{u_3},...,A^{u_m}$ be the state transition ...
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### Policy and Value function as a function of Discount

I'm learning MDP's and I'd like to know if I'm doing these exercises correctly: Consider the following MDP: The agent starts at the cell marked by $S$. Whenever it enters a cell with a positive ...
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### How to learn the value function in a two-player game?

In single-player games, the optimal policy can be derived from the state value function v(s): $$\pi(s) = \underset{a}{\text{argmin}} \sum_{s'} p(s'|a,s)(c(a) + v(s'))$$ where c(a) is the cost of ...
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### Is my derivation of the Bellman equation for $q_{\pi}$ in terms of $p(s'|s,a)$ and $r(s,a)$ correct?

I have done exercise 3.29 from Sutton and Barto and I'd like to check if it's correct. Here's the exercise: Rewrite the Bellman equation for the function $q_{\pi}$ in terms of the three argument ...
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### Policy and Discount Factor

This question is similar to this question, however it has a different question. I'm learning MDP's and I'd like to know if I'm doing these exercises correctly: Consider the following MDP: Suppose a ...
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### Is my derivation of the Bellman equation for $v_{\pi}$ in terms of $p(s'|s,a)$ and $r(s,a)$ correct?

I have exercise 3.29 from Sutton and Barto and I'd like to check if it's correct. Here's the exercise: Rewrite the Bellman equation for the value function $v_{\pi}$ in terms of the three argument ...
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### Rewrite the four Bellman equations for the four value functions $(v_{\pi},v_*,q_{\pi},q_*)$ in terms of $p$ (3.4) and $r$ (3.5) [duplicate]

I have done exercise 3.29 from Sutton and Barto and I'd like to check if it's correct. Here's the exercise: Rewrite the four Bellman equations for the four value functions $(v_{\pi},v_*,q_{\pi},q_*)$ ...
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### Exercise 3.21 Sutton Barto: Draw or describe the contours of the optimal action-value function for putting, $q_{*}(s, putter)$, for the golf example

I am doing exercise 3.21 in Sutton and Barto. Here's the exercise: Draw or describe the contours of the optimal action-value function for putting, $q_{*}(s, putter)$, for the golf example. Here's ...
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### Does maximizing the value function and maximizing the state-action value function generate the same optimal policy?

In reinforcement learning, we define the optimal policy $\pi^*$ as the policy that maximizes the value of the state: $$\pi_v^*=\underset{\pi}{\operatorname{argmax}} {V_{\pi}(s)}$$ In Q-learning, we ...
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### How do we estimate the value of a stochastic policy?

I'm learning about reinforcement learning, particularly policy gradient methods and actor-critic methods. I've noticed that many algortihms use stochastic policies during training (i.e. they select ...
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### In Value Iteration, why can we initialize the value function arbitrarily?

I have not been able to find a good explanation of this, other than statements that the algorithm is guaranteed to converge with arbitrary choices for initial values in each state. Is this something ...
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### PPO custom implementation: do metrics like value loss, actor loss and entropy move a certain way?

I'm wondering whether problems with a custom PPO implementation (problem couldn't be made into a gym environment) can be debugged considering how the losses change over time. In my current experiment, ...
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### Can we also estimate $V_{\pi}$ with SARSA?

For SARSA, I know we can estimate the action value $Q(s,a)$, and the relationship between $V(s)$ and $Q(s,a)$ is $V_{\pi}(s) = \sum_{a \in \mathcal{A}} \pi(a|s)Q_{\pi} (s,a)$. So my question is, can ...
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### What are the recurrences used for updating state value function in $TD$ and $TD(\lambda)$ learning?

There are two types of value functions in reinforcement learning: State value function $V^{\pi} (s)$, state-action value function $Q^{\pi}(s, a)$. State value function: This value tells us how good ...
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### Is there any difference between an objective function and a value function?

I found the usage of both objective function and value function in the same context. Context #1: In the paper titled Generative Adversarial Nets by Ian J. Goodfellow et al. We simultaneously train G ...
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### How can we find the value function by solving a system of linear equations?

I am following the book "Reinforcement Learning: An Introduction" by Richard Sutton and Andrew Barto, and they give an example of a problem for which the value function can be computed ...
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### Is it possible to have values of the states equal to $0$ at the end of the value iteration?

I am new to Reinforcement Learning and I am trying to self learn it. I have already posted some quesiton here and your answershave been really useful to me, so here I am posting another one. I am ...
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### How would I compute the optimal state-action value for a certain state and action?

I am currently trying to learn reinforcement learning and I started with the basic gridworld application. I tried Q-learning with the following parameters: Learning rate = 0.1 Discount factor = 0.95 ...
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### Why do I get the best policy before Q values converge using DQN?

I have implemented DQN algorithm and wonder why during testing, the best performance is achieved by a policy from about 300 episode, when mean Q values converge at about 800 episode? Mean Q-values ... 132 views

### What are the popular approaches to estimating the Q-function?

I need the q-value for my RL training, there are some approaches: Brute-force the action sequence (this won't work for long sequence) Use a classic algorithm to optimise and estimate (this ain't much ...
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### What is a "learned policy" in Q-learning?

I am completing an assignment at the moment. One of the assignment questions asks how you identified the learned policy and how you obtained it. The question is a reinforcement learning question, and ... 1 vote
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### How do I learn the value function for a POMDP with a single-step horizon (bandit)?

Consider a POMDP with a finite number of environment states, $|\mathcal{S}| = N$, but the number of belief states is uncountably infinite. The belief state space is the convex hull of an $N$ simplex. ...
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### How to derive matrix form of the Bellman operators?

Reading the Retrace paper (Safe and efficient off-policy reinforcement learning) I saw they often use a matrix form of the Bellman operators, for example as in the picture below. How do we derive ...
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### What is the Bellman Equation actually telling?

What does the Bellman equation actually say? And are there many flavours of that? I get a little confused when I look for the Bellman equation, because I feel like people are telling slightly ...
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### Equivalence of the $Q(s,a)$ and $V(s)$ under optimality conditions?
Are the state-action values and the state value function equivalent for a given policy? I would assume so as the value function is defined as $V(s)=\sum_a \pi(a|s)Q_{\pi}(s,a)$. If we are operating a ...