Questions tagged [markov-decision-process]

For questions related to the concept of Markov decision process (MDP), which is a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision-maker. The concept of MDP is useful for studying optimization problems solved via dynamic programming and reinforcement learning.

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What is the logic in including/not including subscript $\pi$ in in "E" for value functions? [closed]

Here are two relations for value functions: Eq.1: $v_{\pi}(s)=E_{\pi}[q_{\pi}(S_t, A_t)|S_t=s]$ Eq.1: $q_{\pi}(s,a)=E[R_{t+1}+\gamma v_{\pi}(S_{t+1})|S_t=s, A_t=a]$ Question: Why is there subscript $\...
DSPinfinity's user avatar
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What does the term "expected leaf node" in this exercise from Sutton-Barto mean?

What does the term "expected leaf node" in the Exercise below from Sutton-Barto mean?
DSPinfinity's user avatar
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Derivation of consistency equations for state-value function in Sutton-Barto

I need a step-by step derivation of the consistency equations for state-value function in Sutton-Barto:
DSPinfinity's user avatar
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1 answer
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Is there any example of a Markov Decision Process (MDP) with infinite number of states?

I was learning fundamentals of reinforcement learning from various sources like coursera, Udacity, ...
Bhavesh Achhada's user avatar
1 vote
1 answer
51 views

When is it non-Markovian?

Several months ago, I was writing for class. I claimed an environment was non-Markovian because it would take several states to de-alias some positions in the grid world. I was corrected that it was ...
foreverska's user avatar
1 vote
1 answer
26 views

How to properly model the MDP of a weighted graph with the constraint of only visiting each vertex once (and not get stuck in infinite loops)?

I'm trying to model a MDP to traverse a complete weighted graph (i.e. all vertex are connected). The states, and also the actions (i.e. S=A), are the vertex of the weighted graph. The transition ...
Cristian García Romero's user avatar
1 vote
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Solving MDP as linear program: why minimize the sum of the states' values?

This is a follow-up question to the answer to How can we use linear programming to solve an MDP? Quick recap: the $max$ operators that appear in the Bellman optimality equations can be turned into a ...
Celelibi's user avatar
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Can Q-learning rewards and next states be non-deterministic?

I am working in a team to develop a Q-learning based approach for hyperparameter tuning. I have a disagreement with one of my teammates on how they defined this problem. They defined it as follows: ...
Ahmed Mokhtar's user avatar
1 vote
0 answers
63 views

When can we unnest the minimizations/recursions in an value function(bellman optimality equation)?

When reading the following paper(page 4): An Approximate Dynamic Programming Approach for Dual Stochastic Model Predictive Control I could see that they were able to unnest the minimization's in the ...
richard baws's user avatar
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Optimal decision with continuous, stochastic signals and rewards

I am performing a task, where I have to decide which projects to pursue at a given point in time, where the projects have different horizons of 30 minutes. At a given point in time, forecasts are made ...
Mathman's user avatar
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How can I prove that early termination in an MDP state is valid?

I have an MDP $M$ with transition function $p$, states $S$ and a state $s^0 \in S$. $s^0$ has the property that both the most optimistic trajectory (highest expected reward) and the most pessimistic ...
corazza's user avatar
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Is this a bandit problem or a MDP?

I am trying to understand if this problem can be casted both as a bandit problem as well as an MDP. Lets assume that we are trying to optimize sales $y_t$ based on investments $x_{1, t}, x_{2, t}$ ...
hugh's user avatar
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UCB, Thompson sampling etc seems myopic/greedy for bandits?

When considering multi-armed bandits in different formats, UCB, $\epsilon$-greedy, thompson sampling etc seems so greedy/myopic in the sense that it solely considers reward for the current timestep. ...
hugh's user avatar
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Why is R(s) more restrictive than R(s, a) in an MDP?

I am quite new to RL. I would like to know why a state-dependent reward function R(s) is more restrictive than a state-action-dependent reward function R(s, a)? And why is it that a policy can be ...
TicTacToemat's user avatar
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How to decide on size of state space?

I'm in a reinforcement learning class and we're working on a dice game problem where you profit the number rolled for certain digits and lose all your money on certain other digits. You can choose to ...
faangorn's user avatar
2 votes
1 answer
110 views

MDP to model finding an optimal sequence of actions with no other state data

I would like to build an RL agent who's objective is to find an N length sequence of actions to a reward where N is not known and the states are indistinguishable from each other (at least for it's ...
foreverska's user avatar
2 votes
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179 views

How can i cast this problem as an POMDP or an belief space MDP?

I am currently attempting to formulate my problem as either a Bayes Adaptive MDP or a Belief space MDP. My end goal is to gain insights into the proposed methods within this domain and understand how ...
paul's user avatar
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If the agent is at the same state but at different times and receives a different reward, wouldn't this be violating somehow the MDP assumption?

I've been trying to train an agent, I've received and read suggestions to improve its speed to reach the goal. The suggestion is to use a time penalty, for example, adding $-0.1$ to the reward each ...
Andrea Carolina Mora Lopez's user avatar
1 vote
1 answer
49 views

How is the Markov property of a general state-space model derived?

Below is the derivation for the Markov property of a general state-space model. The red part is not clear. Could someone please explain the steps in the sequential derivation for the red part?
DSPinfinity's user avatar
4 votes
1 answer
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Could you explain these 2 steps of the derivation of the Bellman equation as a recursive equation in Sutton & Barto?

I am reading the Sutton & Barto (2018) RL textbook. On page 59, it derives the recursive property of the state-value function as below. Could you explain the steps of third and fourth equality? ...
tesio's user avatar
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Can we use Low rank approximation in Markov decision process problem?

I am newbie in MDP.I have started reading Ronald Howard Dynamic Programming and MDP book as well as Sutton and Barto An Introduction to Reinforcement Learning. To my understanding MDP is a model based ...
Homer Jay Simpson's user avatar
1 vote
1 answer
173 views

How can I find an upper bound on the number of iterations required to have less than $\varepsilon$ difference in the value of state?

I learned about the Value Iteration algorithm which can help find an optimal policy and values of an MDP with state rewards: $$V_0(s)=R(s)$$ $$V_{t}(s)=R(s)+\gamma\cdot\underset{a}{max}\underset{s'}{\...
Daniel's user avatar
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Convergence of Value Iteration for Discount factor of 1

Given this pseudo code for value iteration: In the case of gamma=1, under what conditions on the MDP will we still be able to find the optimal policy?
Toffe1369's user avatar
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1 answer
583 views

Should DQN/PPO be used for state spaces that are not that large?

I'm interested in trying out Q-learning to solve a problem where I already have a simulation of the environment that can run at about 100,000 fps or steps/sec. Its also continuous with no terminal ...
gameveloster's user avatar
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Does fixing an action in the state transition function of an MDP yield a transition matrix?

I stumbled across this seemingly elementary question while studying the theory of Markov Decision Processes. Suppose $\mathcal{M} = (\mathcal{S}, \mathcal{A}, \mathcal{P}, \mathcal{R})$ is an MDP and ...
Othman El Hammouchi's user avatar
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0 answers
44 views

Would the optimal policy remain same, if I replace R with V*?

In the context of RL, say I'm performing Value Iteration on a reward function R1. And the converged optimal policy is P1 and values are V1. Then, let's say I set rewards to be R2=V1 and perform value ...
famishedrover's user avatar
2 votes
1 answer
152 views

What is the difference between these two versions of the Bellman equation?

The first version is the one I am most familiar with: $$V_\pi(s) = \sum_{a}^{}\pi(a|s) \sum_{s'}^{}T(s, a, s')[R(s, a, s') + \gamma V_\pi(s')]$$ where $T(s, a, s')$ represents the probability of ...
Nova's user avatar
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MDP with a non-markovian reward function?

I have set up a RL environment and it converges to a decent solution when using a reward function: $R(s_t,a_t) = fenv(s_t, a_t)$ , where $fenv$ is the environment dynamics. Now, i want to change the ...
StarDust_08's user avatar
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1 answer
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How can we approximate infinite horizon MDP with finite horizon MDP in the context of reinforcement learning?

For a given value of "discount factor" (and reward values' range) in fixed finite horizon markov decision process (MDP), upto how many episodes we have to extend this MDP so that we can ...
Engr. Moiz Ahmad's user avatar
1 vote
1 answer
161 views

How to correctly evaluate the state value of this simple markov decision process?

For some contexts, I'm working on a c# library for reinforcement learning. I implemented two methods to evaluate a state value function, namely the TD(0) method and the Monte Carlo first visit method. ...
nathan raynal's user avatar
1 vote
0 answers
41 views

How to compare different trajecories in a Markov Decision Process

I realize that my question is a bit fuzzy and I am sorry for that. If needed, I will try to make it more rigorous and precice. Let $\mathcal{M}$ be a Markov Decision Process, with state space $\...
Onil90's user avatar
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2 votes
1 answer
566 views

In the Policy Gradient Theorem proof, why is $d^\pi(s) = \sum_{k=0}^{\infty}\gamma^{k}Pr(s_0 \rightarrow s, k, \pi)$ true?

I was reading the original Policy Gradient Paper. I didn't quiet get the last step of the proof for the policy gradient theorem. The proof given in the paper is below: I don't understand how the last ...
Fady's user avatar
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2 votes
1 answer
316 views

What's the relationship between Bayesian RL and POMDPs?

Bayesian RL seems concerned with having uncertainty over the transition function of the environment, while POMDPs try to capture uncertainty over the state one is currently in. However, both end up ...
mdc's user avatar
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1 answer
719 views

Markov's Decision Process - calculate value in each iteration

I have the following decision tree: I calculated the value of the plan using the following paramenters (given): {𝑆0 β†’ π‘Ž1 , 𝑆1 β†’ π‘Ž3 , 𝑆2 β†’ π‘Ž4 }, Discount factor (𝛾)= 0.2 I used this formula to ...
stuckincode's user avatar
1 vote
1 answer
85 views

How to prove that an action-value function optimal for one problem formulation is also optimal for another?

I want to ask about the intuition/where-to-look/what-to-try if I want to prove that an action value function optimal for a problem is also optimal for another reformulation of that smae problem. For ...
Souhaielrmx's user avatar
2 votes
1 answer
172 views

Remove already reached targets from the system to enable reaching other targets?

This may be a very fundamental question, but somehow I can't decide. I have a graph and the user can take several actions while traversing it and there are multiple points with rewards. When I execute ...
Ferda-Ozdemir-Sonmez's user avatar
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0 answers
53 views

How to represent the state and action space when modeling card game Love Letter using Markov Decision Process

Now my task is to use model checking to find a best strategy for card game Love Letter, and I need to model the game using Markov Decison Process first. I have done a lot of resaerch and I decided to ...
jiaxin886's user avatar
4 votes
0 answers
171 views

What is the correct interpretation of the discount factor in MDPs?

In infinite-horizon MDPs one can consider the expected discounted return from a distribution of start states as the objective[^1]. i.e. $\mathbb{E}[V^{\pi}(S_0)] = \mathbb{E}[G_0] = \mathbb{E}[\sum_{t=...
Skander's user avatar
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0 answers
33 views

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 $(v_{\pi},v_*,q_{\pi},q_*)$ ...
user's user avatar
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1 vote
2 answers
544 views

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 ...
Cloudy's user avatar
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$E_{\pi}[R_{t+1}|S_t=s,A_t=a] = E[R_{t+1}|S_t=s,A_t=a]$?

I would like to solve the first question of Exercise 3.19 from Sutton and Barto: Exercise 3.19 The value of an action, $q_{\pi}(s, a)$, depends on the expected next reward and the expected sum of the ...
user's user avatar
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1 vote
0 answers
53 views

Determining a policy to play a game of chance

I'm trying to optimize the expected return from a game of chance, but have quickly realized the problem outclasses the introductory AI course I took in college years ago. I would appreciate any ...
user avatar
1 vote
0 answers
48 views

Number of possible joint policies in a Dec-POMDP and the time required to evaluate each one

I was reading a book about Dec-POMDPs and came across this curious result where the author specifies the number of possible joint policies to evaluate and the time needed to evaluate a single joint ...
Souhaielrmx's user avatar
1 vote
1 answer
269 views

Can the state transition function be dynamic in reinforcement learning?

In general, there are two types of transition functions in reinforcement learning. Mathematically, they are as follows #1: Stochastic state transition function: $$T : S \times A \times S \rightarrow [...
hanugm's user avatar
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2 votes
1 answer
211 views

Is it appropriate to represent 'total failure' as an absorbing state?

My understanding is that, in Markov decision processes, absorbing state are states which can transition only to themselves and that these transitions generate rewards of 0. I know that absorbing ...
K--'s user avatar
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Does the policy search work if there is no state to state dependency through actions?

There is a game in which the state comes one after the other without depending on the agent's action. The agent gets a reward for its actions at the end of the game. The goal of the agent is to reach ...
veerendra's user avatar
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1 answer
73 views

Does $S_{t+1}$ denote the future information in Q-learning?

In Q-learning, $Q(S_t,a)$ is updated by the Bellman equation. $Q(S_t,a) = r + \max_{a'}(Q(S_{t+1},a'))$ where $S_{t+1}$ is the future state. Let's say $S$ denotes the stock price, does it mean we are ...
L.Chau's user avatar
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Does $R_{s}=E[R_{t}|S_{t}=s]$ indicate the reward we might expect on getting on average moving from any other state to $s$?

I'm trying to understand correctly what each "variable" in RL is and I'm not sure about $R_{s}$ the reward function. I used to think that it's the reward we may expect on average after ...
Daviiid's user avatar
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2 votes
1 answer
785 views

Factors that affect the number of iterations of value iteration

I had an assumption that value iteration will take more iterations to converge if the map size increases/environment's complexity increases. I tried to verify this idea by running value iteration on ...
john li's user avatar
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1 vote
1 answer
191 views

Are these two forms of the state value function the same?

Why are there different forms of the value function in reinforcement learning? Sutton & Barto (2nd edition, equation 3.14) define the state value function as follows $$v_{\pi}(s) = \displaystyle\...
cgo's user avatar
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