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Markov Decision Processes Verified Solutions

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Markov Decision Processes Verified Solutions Markov decision processes ️️MDP - formally describe an environment for reinforcement learning - environment is fully observable - current state completely characterizes the process - Almost all RL problems can be formalised as MDP - optimal con...

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  • October 30, 2024
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  • 2024/2025
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Markov Decision Processes Verified Solutions

Markov decision processes ✔️✔️MDP - formally describe an environment for reinforcement learning

- environment is fully observable

- current state completely characterizes the process

- Almost all RL problems can be formalised as MDP

- optimal control primarily deals with continuous MDPs

- Partially observable problems can be converted into MDPs

- Bandits are MDPs with one state



Markov Property ✔️✔️- future is independent of the past given the present

-the state captures all relevant information from the history

- once the state is known the history can be thrown away

- the state is a sufficient statistic of the future



State transition Matrix ✔️✔️- markov state s and successor state s', the state transition probability

- state transition matrix P defines transition probabilities from all states s to all successor states s'



Markov Process ✔️✔️- markov process is a memoryless random process i.e, a sequence of random
states S1, S2... with the markov property

-Markov process (or Markov Chain) is a tuple <S,P>

- S is a (finite) set of states

- P is a state transition probability matrix



Markov reward process ✔️✔️- A markov reward process is a Markov Chain with values

- Markov reward process is a tuple <S,P,R,Y>

- S is a finite set of a states

- P is a state transition probability matrix

, - R is a reward function

-Y is a discount factor



Return ✔️✔️- Return Gt is the total discounted reward from time-step t

- the discount Y is the present value of future rewards

- value of receiving reward R after k+1 time-steps is Y^k R

- values immediate reward above delayed reward

- y lose to 0 leads to "myopic" evaluation

- y close to 1 leads to "far sighted" evaluation



Discount ✔️✔️- mathematically convenient to discount rewards

- Avoids infinite returns in cyclic Markov Processes

- Uncertainty about the future may not be fully represented

- if reward is financial, immediate rewards may earn more interest than delayed rewards

- animal/human behavior shows preference for immediate reward

- sometimes possible to use undiscounted Markov reward processes if all sequences terminate



Value Function ✔️✔️-Value function v(s) gives the long-term value of state s

- state value function v(s) of an MRP is the expected return starting from state s



Bellman Equation for MRPs ✔️✔️the value function can be decomposed into two parts:

- immediate reward Rt+1

- discounted value of successor state Yv(St+1)



Bellman Equation in Matrix Form ✔️✔️- Bellman equation can be expressed concisely using matrices,

v=R+yPv

v is a column vector with on entry per state

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