# 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 ${y}^{\left(i\right)}$ ).

what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!