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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

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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