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Or, when we are writing rewards as a function of the states only, this becomes

R ( s 0 ) + γ R ( s 1 ) + γ 2 R ( s 2 ) + .

For most of our development, we will use the simpler state-rewards R ( s ) , though the generalization to state-action rewards R ( s , a ) offers no special difficulties.

Our goal in reinforcement learning is to choose actions over time so as to maximize the expected value of the total payoff:

E R ( s 0 ) + γ R ( s 1 ) + γ 2 R ( s 2 ) +

Note that the reward at timestep t is discounted by a factor of γ t . Thus, to make this expectation large, we would like to accrue positive rewardsas soon as possible (and postpone negative rewards as long as possible). In economic applications where R ( · ) is the amount of money made, γ also has a natural interpretation in terms of the interest rate (where a dollar today isworth more than a dollar tomorrow).

A policy is any function π : S A mapping from the states to the actions. We say that we are executing some policy π if, whenever we are in state s , we take action a = π ( s ) . We also define the value function for a policy π according to

V π ( s ) = E R ( s 0 ) + γ R ( s 1 ) + γ 2 R ( s 2 ) + s 0 = s , π ] .

V π ( s ) is simply the expected sum of discounted rewards upon starting in state s , and taking actions according to π . This notation in which we condition on π isn't technically correct because π isn't a random variable, but this is quite standard in the literature.

Given a fixed policy π , its value function V π satisfies the Bellman equations :

V π ( s ) = R ( s ) + γ s ' S P s π ( s ) ( s ' ) V π ( s ' ) .

This says that the expected sum of discounted rewards V π ( s ) for starting in s consists of two terms: First, the immediate reward R ( s ) that we get rightaway simply for starting in state s , and second, the expected sum of future discounted rewards. Examining the second termin more detail, we see that the summation term above can be rewritten E s ' P s π ( s ) [ V π ( s ' ) ] . This is the expected sum of discounted rewards for starting in state s ' , where s ' is distributed according P s π ( s ) , which is the distribution over where we will end up after taking the first action π ( s ) in the MDP from state s . Thus, the second term above gives the expected sum of discounted rewardsobtained after the first step in the MDP.

Bellman's equations can be used to efficiently solve for V π . Specifically, in a finite-state MDP ( | S | < ), we can write down one such equation for V π ( s ) for every state s . This gives us a set of | S | linear equations in | S | variables (the unknown V π ( s ) 's, one for each state), which can be efficiently solved for the V π ( s ) 's.

We also define the optimal value function according to

V * ( s ) = max π V π ( s ) .

In other words, this is the best possible expected sum of discounted rewards that can be attained using any policy. There is also a version of Bellman'sequations for the optimal value function:

V * ( s ) = R ( s ) + max a A γ s ' S P s a ( s ' ) V * ( s ' ) .

The first term above is the immediate reward as before. The second term is the maximum over all actions a of the expected future sum of discounted rewards we'll get upon after action a . You should make sure you understand this equation and see why it makes sense.

We also define a policy π * : S A as follows:

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
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Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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