# 2.12 Machine learning lecture 12 course notes  (Page 2/8)

 Page 2 / 8

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

$R\left({s}_{0}\right)+\gamma R\left({s}_{1}\right)+{\gamma }^{2}R\left({s}_{2}\right)+\cdots .$

For most of our development, we will use the simpler state-rewards $R\left(s\right)$ , though the generalization to state-action rewards $R\left(s,a\right)$ 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:

$\mathrm{E}\left[R,\left({s}_{0}\right),+,\gamma ,R,\left({s}_{1}\right),+,{\gamma }^{2},R,\left({s}_{2}\right),+,\cdots \right]$

Note that the reward at timestep $t$ is discounted by a factor of ${\gamma }^{t}$ . 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 $R\left(·\right)$ is the amount of money made, $\gamma$ 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 $\pi :S↦A$ mapping from the states to the actions. We say that we are executing some policy $\pi$ if, whenever we are in state $s$ , we take action $a=\pi \left(s\right)$ . We also define the value function for a policy $\pi$ according to

${V}^{\pi }\left(s\right)=\mathrm{E}\left[R,\left({s}_{0}\right),+,\gamma ,R,\left({s}_{1}\right),+,{\gamma }^{2},R,\left({s}_{2}\right),+,\cdots \right|{s}_{0}=s,\pi \right].$

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

Given a fixed policy $\pi$ , its value function ${V}^{\pi }$ satisfies the Bellman equations :

${V}^{\pi }\left(s\right)=R\left(s\right)+\gamma \sum _{{s}^{\text{'}}\in S}{P}_{s\pi \left(s\right)}\left({s}^{\text{'}}\right){V}^{\pi }\left({s}^{\text{'}}\right).$

This says that the expected sum of discounted rewards ${V}^{\pi }\left(s\right)$ for starting in $s$ consists of two terms: First, the immediate reward $R\left(s\right)$ 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 ${\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{s\pi \left(s\right)}}\left[{V}^{\pi }\left({s}^{\text{'}}\right)\right]$ . This is the expected sum of discounted rewards for starting in state ${s}^{\text{'}}$ , where ${s}^{\text{'}}$ is distributed according ${P}_{s\pi \left(s\right)}$ , which is the distribution over where we will end up after taking the first action $\pi \left(s\right)$ 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 ${V}^{\pi }$ . Specifically, in a finite-state MDP ( $|S|<\infty$ ), we can write down one such equation for ${V}^{\pi }\left(s\right)$ for every state $s$ . This gives us a set of $|S|$ linear equations in $|S|$ variables (the unknown ${V}^{\pi }\left(s\right)$ 's, one for each state), which can be efficiently solved for the ${V}^{\pi }\left(s\right)$ 's.

We also define the optimal value function according to

${V}^{*}\left(s\right)=\underset{\pi }{max}{V}^{\pi }\left(s\right).$

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:

${V}^{*}\left(s\right)=R\left(s\right)+\underset{a\in A}{max}\phantom{\rule{1pt}{0ex}}\gamma \sum _{{s}^{\text{'}}\in S}{P}_{sa}\left({s}^{\text{'}}\right){V}^{*}\left({s}^{\text{'}}\right).$

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 ${\pi }^{*}:S↦A$ as follows:

#### Questions & Answers

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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