# 2.12 Machine learning lecture 12 course notes  (Page 7/8)

 Page 7 / 8

For example, one may choose to learn a linear model of the form

${s}_{t+1}=A{s}_{t}+B{a}_{t},$

using an algorithm similar to linear regression. Here, the parameters of the model are the matrices $A$ and $B$ , and we can estimate them using the data collected from our $m$ trials, by picking

$arg\underset{A,B}{min}\sum _{i=1}^{m}\sum _{t=0}^{T-1}{∥{s}_{t+1}^{\left(i\right)},-,\left(A,{s}_{t}^{\left(i\right)},+,B,{a}_{t}^{\left(i\right)}\right)∥}^{2}.$

(This corresponds to the maximum likelihood estimate of the parameters.)

Having learned $A$ and $B$ , one option is to build a deterministic model, in which given an input ${s}_{t}$ and ${a}_{t}$ , the output ${s}_{t+1}$ is exactly determined. Specifically, we always compute ${s}_{t+1}$ according to Equation  [link] . Alternatively, we may also build a stochastic model, in which ${s}_{t+1}$ is a random function of the inputs, by modelling it as

${s}_{t+1}=A{s}_{t}+B{a}_{t}+{ϵ}_{t},$

where here ${ϵ}_{t}$ is a noise term, usually modeled as ${ϵ}_{t}\sim \mathcal{N}\left(0,\Sigma \right)$ . (The covariance matrix $\Sigma$ can also be estimated from data in a straightforward way.)

Here, we've written the next-state ${s}_{t+1}$ as a linear function of the current state and action; but of course, non-linear functions are also possible.Specifically, one can learn a model ${s}_{t+1}=A{\phi }_{s}\left({s}_{t}\right)+B{\phi }_{a}\left({a}_{t}\right)$ , where ${\phi }_{s}$ and ${\phi }_{a}$ are some non-linear feature mappings of the states and actions. Alternatively, one can also use non-linear learning algorithms, such as locally weighted linearregression, to learn to estimate ${s}_{t+1}$ as a function of ${s}_{t}$ and ${a}_{t}$ . These approaches can also be used to build either deterministic or stochastic simulatorsof an MDP.

## Fitted value iteration

We now describe the fitted value iteration algorithm for approximating the value function of a continuous state MDP. In the sequel, we will assumethat the problem has a continuous state space $S={\mathbb{R}}^{n}$ , but that the action space $A$ is small and discrete. In practice, most MDPs have much smaller action spaces than state spaces. E.g., a car has a 6d state space, and a2d action space (steering and velocity controls); the inverted pendulum has a 4d state space, and a 1d action space; a helicopter has a 12d state space, and a4d action space. So, discretizing ths set of actions is usually less of a problem than discretizing the state space would have been.

Recall that in value iteration, we would like to perform the update

$\begin{array}{ccc}\hfill V\left(s\right)& :=& R\left(s\right)+\gamma \underset{a}{max}{\int }_{{s}^{\text{'}}}{P}_{sa}\left({s}^{\text{'}}\right)V\left({s}^{\text{'}}\right)d{s}^{\text{'}}\hfill \\ & =& R\left(s\right)+\gamma \underset{a}{max}{\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{sa}}\left[V\left({s}^{\text{'}}\right)\right]\hfill \end{array}$

(In "Value iteration and policy iteration" , we had written the value iteration update with a summation $V\left(s\right):=R\left(s\right)+\gamma {max}_{a}{\sum }_{{s}^{\text{'}}}{P}_{sa}\left({s}^{\text{'}}\right)V\left({s}^{\text{'}}\right)$ rather than an integral over states; the new notation reflects that we are now working in continuous states rather than discrete states.)

The main idea of fitted value iteration is that we are going to approximately carry out this step, over a finite sampleof states ${s}^{\left(1\right)},...,{s}^{\left(m\right)}$ . Specifically, we will use a supervised learning algorithm—linear regression in our description below—to approximate the value functionas a linear or non-linear function of the states:

$V\left(s\right)={\theta }^{T}\phi \left(s\right).$

Here, $\phi$ is some appropriate feature mapping of the states.

For each state $s$ in our finite sample of $m$ states, fitted value iteration will first compute a quantity ${y}^{\left(i\right)}$ , which will be our approximation to $R\left(s\right)+\gamma {max}_{a}{\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{sa}}\left[V\left({s}^{\text{'}}\right)\right]$ (the right hand side of Equation  [link] ). Then, it will apply a supervised learning algorithm to try to get $V\left(s\right)$ close to $R\left(s\right)+\gamma {max}_{a}{\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{sa}}\left[V\left({s}^{\text{'}}\right)\right]$ (or, in other words, to try to get $V\left(s\right)$ close to ${y}^{\left(i\right)}$ ).

#### Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
learn
Google
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
Hafiz Reply
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
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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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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