But the unknown parameter is.

Still the vector of
regression go fissions.

So the number of parameters is be and
it's going to be large so

you have to do something to do with
high dimensional view of the problem.

OK then the last so it's like this.

So you Dake the many miser
of the following you dick to

the sum of squares I normalized by N.

that's my habit you don't have to OK And

then there's a penalty which
is proportional to the L.

one norm of the Kofi shits.

And I put a two here because
otherwise I have to right

to send it out of place OK so
I guess this procedure is known and

maybe you worked on this theory or
plighted.

OK so now I want to show a but
so I talked about.

A fix or sparsity on Friday but maybe you
were not there but anyway I want to show

its goal a little bit into depth topic so
you need a certain condition

to get good theoretical properties of
the system mater in that condition.

I call this compatibility but it's
related to restrict that I can fell use.

And I want to you really to show you that
this condition is really quite natural So

that's why I am going to
do you write it here and

really big to make life simple
let's forget about the noise for

a moment OK so
I'm going to cancel the noice.

Just just for development of the series

later on I'll show you that's quite
useful to think about that case too.

So if you have no noise I
have this kind of thing so

there's no noise so why is just X.

beat us zero so you don't you get X.

The mind is the zero squared.

One.

OK So let's consider this problem
first there's no noise now.

OK now I'm going to write down
the derivative of booted to Ciro and

it's called the K K T conditions while.

Conditions this is taking that the
reflective of this term that's extra and

spose X.

B..

Minus Zero and then a two but
there's a two and

zero to ten PSI too so I'm just cancelling
the two digits the derivative of this term

and then the reflective of the this
term is to sup differential

call it sit so
you put the derivative to zero and

solve maybe I should get
a star here made it spit so

this is the sub differential of
the absolute value function or L.

one norm.

So.

This says that set this effect or

which components are the sign
the sign off beat a star and

if beat a star is Ciro I
don't care what set this but

it's something between minus one and
one day so sit.

The largest value in society so that's
an absolute fall you list in one OK.

OK now I'm going to multiply by
because Star Mine has become

so what I get then if I multiply
here by beat a Star Mine is beat or

transpose so you get again this quantity.

X.

B.

star minus zero squared.

And then I multiply by beat a star minus
beat on this side too so this is lumped.

Be a star minus zero so far so good.

OK so now we have let me just write it so
it's beat.

To true beat transpose zip star minus and
here

and then beat a star transpose it just OK.

OK now I'm going to use that the super
norm of said Star is less than one so

I can bound disturbed by the L.

one normal B.

just zero OK And
this storm is exactly the L.

one normal fee to start OK.

Now we're almost done and
then then we get to a result so

let's see let me call this zero
ective set of zero so that's the.

Set were to go fishes are nonzero
you can be more precise here but

let's not do that we just
look at strong sparsity.

At zero is going to be the cardinality
of that settle dispersed city.

And I'm going to write B..

Or maybe B.

maybe better B.

This zero is going to be B.

J..

G.

in this zero.

And B.

minus zero so I'm just going to split.

Perfecter according to being
in the active set or not.

And let's see them again right here.

And so this is to sing right.

And here I can split it up so you get.

OK minus love.

There just split it up and
inside and outside perspective said.

And here I used to triangle
inequality we're almost done OK love.

B..

This is zero minus the.

Zero just.

So.

And now comes to the trick
let me put it here.

I'm going to define
the following quantity.

I put a head here don't worry
about that squared minimum.

Be square search that beats.

The zero is equal to one

and I normalize so times as you like this.

So this is the crucial quantity This is
called a competent billet a constant so

it's just introduced to
make the following step.

OK So if this is so
negative I'm done then I just proved that

the prediction error for the last so
is less than or equal to zero and

then I'm done so it Ciro so then so I can
that's not an interesting case of what if

this is positive news that this
storm is larger than depth or and

then with this normalization
you can bounded by

love and then because of
the there are no squares here so

there's no square in
that split this queer So

there's this OK I'm putting
to square here a couple.

Closer it's just a couple of stone stairs.

And this is just.

This.

I've pulled ahead here
because of the Random Act of

sixty nine so that's step one times X..

This year will no square here so
now it's still without a square so here.

Yeah so

if you are OK just defining this
funny quantity you get a bound for

the prediction error of the last or so
disturbed and cancels with that term and

then you get X.

The Star lines to zero

this and over let's flip the squared and
you get and

don't like square root and
here's this book to square so

you just put it to the other side
you get lumped the squared zero.

Squared this series so
that's a bonus for the prediction error.

And I want to show you know I'm going
to dislike that it's also the so

it's an upper bound and
it's also the lower bound.

OK that's.

OK that's just too good to
make that the last step.

And so this guy had this just depending
on how did on the design its own property

of the design and it's a curse here
explains a little bit further so

that's the main thing this is
a competent at the constant.

It's the point.

Constance.

So the idea is if you look in the
literature similar are going to disappear

but they're occult restricted I can
value so this one is a little bit

different a little bit less stringent
which allows me also to get

the lower bound because actually what
appears in the Indus upper bound

also appears in the lower bound to
Apparently that's the main quantity that.

Rule still being so let me try to.

Give the details known to slight So
yeah sometimes it gets a bit technical and

then I well maybe all the time so
then you're just zoom out OK.

Yes.

This one.

OK yeah you have to think a little bit
because you know if you're if you define

it in this way it's almost by definition
like that it's sort of the definition.

So I don't write it all that it's
not difficult you know it's just.

A whole.

Yes subject to subject very
good question thinks it's here.

OK So we start again so
here's to the linear model again.

Unknown recursion functions and
I'm going to look at lower bounds and

upper banks but mainly I want to show
something new so lower bounds and

let's just for lower bounds to
issue noise Gaussian noise with

the Unitarians is maybe
you starting point B.

parameters and observations and
it's higher dimensional So

you're still also Yeah that's the I
guess most of you are familiar with it.

And also the motivation so I'm not
giving that if you want to ask me the.

Scoop motivation to do it this
way if you think of sparsity.

And the tuning parameter that's something
to remember it's usually a Ford or

square would look over and.

OK this will occur from time to time so

this is to actually said I've written that
on the board the size of the exit is to

sparsity which is sure to
be small whatever it needs.

OK And like I said on the Friday So

typically if you look at
the prediction average at this

difficult it behaves like the number
of parameters defied by the number for

proficiency that the number
of active parameters times

the price to pay for not knowing it OK So

typically it behaves like this so

if I take a lock be defied by end it's
moralist the joining parameter squared so

it's a serial time slot the squared so
you see it appear here.

S.
zero times up the square but there's a.

Constant in the autumn there.

Due to yes some kind of
identifying ability conditions.

OK So actually.

OK so I want to show that this
compatibility constant which

actually appears there is really needed so

I want to establish tight bounds for
the last song for the prediction error for

a start and you can ask why well
the main reason to do look at

type bounce is to show that these kind of
compatibility constants are be necessary.

Well uncertain on Thursday I'll talk
a little both about to the bias law so

in there you will have
the situation where you do a lot so

when actually the true
parameter is not sparse and

then you maybe want to know what happens
and this is largely still open problem

it's all talk about Officer state and
it's nice mathematics I think.

Some geometry I'll talk about
it your materials probably

OK now OK just put down the main
result and there are some conditions

which are off to form the true underlying
parameter if it's not zero then it's not

going to be below the noise level and
up that's called Better mean condition.

So a better min condition
is off to form like this so

the noise level is Norris
squared look below for N.

if you go below this level you cannot
really tell whether the parameters zero or

whether it's small but non-zero so

if you're away from that
that's like these so

there's a constant here so it's a larger
ordered and squared look be over and.

And I'll need such conditions
now as a statistician you

don't want to be shewn that if something
is non-zero then it's large enough so

that you will notice it that's
what this is actually Schumi So

this is a very bad condition from
a statistical point of view so

usually I don't assume that for
lower bounds it's I think OK.

So for computer science they're OK Also as
well they always assume this because they

say if you don't assume that then I can't
find an active set with first at stations

I mean we can discuss this but
if you're honest you don't assume that for

inference but for
lower bounds I think it's OK so

in contrast to proponents because this
is just a condition to make life easy.

OK.

Then I come to discount but ability
constant so I'm not just that's why I

put the head there I'm going
to look at random design and

I take the expectation of the ground
matrix so it's now a deterministic B.

times the Matrix and I'm going to consider
that quantity which I wrote on the board.

But now replacing the.

Cigar matrix by its expectation so
if you're friend of the scientists would

be a random quantity but now I just
have a deterministic quantity and

I call that gone but
to beat it to Constance So

that's a very funny thing it's
you have the courage of etic.

Function here in which you want to
minimize subject to a constraint and

this is a non-convex constraint so
you end up with.

Non conflicts up to my station problem.

Actually what I'll just tell you now is
I'll give you an example later on I hope

it's time where I can explicitly
calculate that thing but

in general it's not so easy.

OK So this is the best I'm in
condition that I'm going to issue so

it's on but it's on the slide let's don't
worry about it like this I mean maybe you

like me like to look at it with just
that I've shown you it that there's some

condition there and then comes
following result OK so you have to.

Say your design is random and say its
to columns or the rows are ID copies

of some factor with some go Ferencz matrix
Sigma Ciro it think of it like that and

to normalize it I just issue
say the diagonal matrix

is old once or are at least
bounded by something i mean i'm

just have to somehow normal isolette
say that all entries are bounded by one

and then that this is important this is
not a normalization I'm going to be sure

that this largest I can sell you
then is not going to grow too fast

so this makes my kinfolk addition is also
occurring very much in the literature so

now comes the result this is a tight Bonet
So should these better mean condition.

Goshen design with the GO Ferentz

matrix with the largest I can fill you or
small order look be.

So difficulty people
shouldn't abounded but

let's allow to grow a little bit
the usual value of the tuning parameter.

The usual assumption on the sparsity and

that's the usual Because you
know if you look at this.

Yes you wanted to go to zero right so

love the square dots lock
squared lock be over N.

times is zero you want it to go to zero so
that's what I'm sure mean here you want

some kind of virgins OK So
that's this condition and

there's some some little technical
conditions here some convergence so

also if a serial is
a little bit larger OK And

then the prediction error off the last so

is this bound you see that it's
just exactly like here Noni

quality here of course not exactly
it's as some thought experience so

with a small order term coming there.

Yeah so this says you can't
get away from discomfort

ellipticals it's just if you
expect me to type bonds.

Yeah so I'll talk about the that one
too but it's more complicated there.

Yes.

Yes so I have to explain a few
things to better conditions I don't

maybe not to go too much
into old a technicality So

let's first just explain discount
the ability little bit further.

So this is my notation I have written
that on the board I do it again.

So to if you have an arbitrary
subset just put everything outside

the subset to zero and in this Victor and
reef leave the rest untouched and

that's called this you can do that for
an arbitrary factor and B.

minus this is to the complement
operation you know.

OK now we're going to do some linear
algebra if you have a matrix you see

metric positive semi definite matrix you
can defined it's minimal I can sell you.

And if TO ME KNOW I can feel you
suppose if you can identify defects or

be from from discourage erratic form OK so

it's Gleick positive gives
you some identify ability

now we're in a situation where you have
an extra set in a non active set and

then you can define If
you have two sub spaces

of linear combinations of the columns
of linear combinations of

variables in essence variables not in this
you can look at its canonical correlation.

Yeah here's a be missing
Thanks yes there's a B.

So yes the canonical correlation
is defined like this

well canonical correlation so the yeah.

It's the angle and
if you have an angle between two vectors

you can also look at their distance and
the distances than one minus that angle

if they're normalized to be
equal to full length one.

Yeah.

Well.

Yes.

I maybe I did the nomination wrong or
that's could be but

I'm just writing this for
two fixed source X.

in and and next to if they have length

one it's two minus their
inner product right.

If they have length one so that.

Then thinking in that direction will go.

Yeah.

Maybe the one half should be here he.

Will.

Have you know the difference
of those two factors so B..

That C.
B.

is B.

This.

Plus B.

minus S.

right well or needed minus whatever.

So.

Now we go to the L.

one world and you can write sed sites
right on similar things in one world so

you just look at the minimal I can value
or I think you'd have changed the notation

here but do you just look at the minimal I
can for you but instead of the restriction

that the L two norm is equal to one you
restrict a one norm to be equal to one

and maybe the one half is
on the wrong place again.

You can also look at some kind of L.

one canonical correlation so

again I forgot to be here with I
replaced the norm here by one more.

You know and then there is
this normal I say sion with S.

steps because if you go from L.

to Norm to a one on you say OK if you
have active sit with this then the norm

is a square root this is a bounded by
square root is time still to Norm so

it makes sense to normalize
with the size because

the normal something is typically
much larger than the two norm so

you'd need some normalization
if you want to compare.

So but the bigger to go and
so if you look at the old

all to the one you find in the literature
compatibility constant it's something like

Canonical correlation
as you replace the L.

to Norm back to L.

one Norm And here's an inequality sign and

there's not one but maybe some other
constant it's something like L.

one type of canonical correlation you
just compare linear combinations in one

space with Lee in your communication
to the other space and

then you look how far
are they from each other.

And this is the new one
which I've written here.

Which gets even more complicated so
it's like here.

Except that you instead of this
restriction you have this restriction.

OK And it turns out that's the real thing
that place a roll in the lower bounds.

And so it's a little bit if you
look at this it's a little bit so

if you put your I think one of them if
you put somewhere if you change a little

bit that definition you get to restrict it
I can tell you but you might also know.

You a smaller one gets
difficult smaller and one.

Very good so you want it to be close to
one because then you should trekked more

and then that helps makes the you wanted
to be large because you're dividing by it

and if this is large if this is one you're
happy this is what I'm doing here but

you have to make it a little bit smaller
than one if you are if you have to do

with Noyce but
depending on the situation or

you know OK so this is to you.

But the I can give you a geometric
interpretation there that you just

see you have here.

This restriction make sure that the first
part is you take the conflicts who

both signed go fix will of
the variables in this and

you look at the assigned convicts
who have the variables and

not in this video maybe use
blow it up a little bit.

And you look how far they are from
each other that's what this is and

then you get this picture so this is just
the active set is just one fairy ball and

all the other ferryboats you
take the sign convicts who and

then you look at this distance and that's
the bolt to be that the constant but

now it's this does new one is
a bit more complicated as it were

difficult interpretation OK so

the new one is bigger than the old
one because well so that means it's.

You may have gained something
because you're dividing by it

increasing in units true and you want to.

You to be.

Increasing you that.

You want you to be large you know and

if you can take it equal to
one you're happy you're so

we are we take it you know
you here there we take one.

So that's now we understand
compatibility and.

So this is we know that the upper bound
depends on the competent beauty constant

we can prove that under the conditions
of that sandwich I started with

you have these upper bound and
for that you do not need.

Any better mint condition so you don't
need to non-zero stay away from zero.

This is actually a new result and
this is the result in literature

which is a little bit you know
here you have nice remainder terms

which are nice and
here is just order of magnitude and

there's tissue and so on so it's
a little bit cleaning up the situation

from the literature OK.

But see I think I'll just skip this part

except this part so.

If you look at this current none
can fix up to my station problem

what is important is to show
that the solution will have

non-serious inside to sit this and
let me just give you.

The geometry of that it's kind
of nice I think three feel that.

So here's to L.

one ball OK here Ciro maybe you know

this picture to let me try to write it
all it is just a picture in the book of.

Sheep should run the hasty and so on so

here if you beat us heroes here and
you want to minimize.

The L.

to distance subject to L.

one norm being equal to one
then you hit a point and

you get to spar solution right
that's that's the idea of the law so

it's we're in a different situation
here still one ball again and

now I want well forget about this
minus a spark just put it to zero for

a moment then I want to minimize.

The the ball subject to the elbow
norm being equal to one so

what I'm actually doing is.

Let me see.

I forgot how it goes with the idea
is you're sort of trying to

blow up a balloon here right

and if you do put a balloon in
a box it will never go into corners

right no matter how hard
you try not my balloons so

it will not be sparse it will not
be a spar solution yes it's more or

less like you do a balloon outside
it until it's inside to the L.

one ball and then it will not be in the
corners and you have to do a little bit of

arguments here because it's in high
dimension you never know what the turns

out to be like that has so you get
a non-zero solution which is kind of good.

For the theory though
let me Schiphol this is.

OK.

I don't know about time let's see.

I'm assuming this quantity is going to
see a row and that's a little bit more of

an issue meaning that you have
a convergence in the first place so

shouldn't call for chance for
a little bit larger set as well OK see.

And then for the noiseless laso and
it's like there but now for normal four to

grow matrix replies replaced by its
expectation you get the following results.

So I'm comparing out a noisy
also with the noiseless less so

take your tuning parameter
of order square o'clock P.

over and here's a quantity which is going
to zero that's why I put it in green.

And then with large probability
the prediction air over the last so

if you compare it with the prediction
arrows to noiselessly also

it's the remainder that
the difference is a smaller order so

different result says you might as well
look at the noiseless also because

the bias dominates to ferry and and
you can think of this is to bias.

And so.

What the remains is a small order.

OK So

this difference is smaller
ordered in the quantities

itself OK I'm just going
to forget about it's

smaller smaller order so
that So this is like

the noise is also due prediction error
of the noise has lost its like the bias.

And if you then compare the noisy loss A
with the noiselessly also disliked a fair

use and that serum which I
just shows you says that this

storm is of Larcher ordered and disturbed
OK so you can might as well look at.

The noise is also to get tight
bounce because that's the main term

OK S.O.S. And
both are clearly these two are equivalent.

God this is a little bit about
the kind of theory you used

to prove this kind of research.

Nothing very new.

And then once you know that you only
have to study to noiseless laso

the prediction error of
the noise of this law so

you can find the expected result and there
but then you need to see these bit I mean

conditions OK so that's where
you where you get to type out.

So for a.

Chick time.

Yes.

Dear.

Thanks.

So to proof this this last
theorem you need some.

Help to my station.

And if you need this I
really need very much that

the solution will have none zeros here if
you can't blow up a balloon in a box and

get all get to the edge to the corners.

If you combine So you combine the two
C.M.'s you get the lower bound for

the noisy losses OK OK.

So this is all about random design and
now let's go to fix design.

So we considered Gaussians
the sine what about sixty seven.

Again let's start with the noiseless case.

So noiselessly also is the one
which was on the board now but

now with here really the goal matrix and
not its expectations so

the noise so this is now a fixed matrix so
it's everything deterministic here.

And then you have to change nothing
you can apply to previous theorem it's

just nothing changed it's
just a different station so

you can get the same result but now.

The definition is with with this matrix
It's just a different matrix there OK.

So for the Norse risqué so
we are done so let's consider an example

an example is for
to go to fairly ation penalty

so let's consider a function well
let's say a function evaluated at and

points so that its effect on
an dimensional fixed or and

then go to fairly Haitian is just the sum
of the absolute value of the difference

if each jumps all our largest this whole
artist is an art and so on still too.

For a function only evaluated and points.

And do so with total variation
penalty still noise says so

I mean in my eyes I have a fixed function

which I don't know maybe well it's it's
not it's a bit artificial it's noice

the situation I want to recover
this function I do the.

Residual sum of squares while
the sum of squares hold a L.

to distance and go to for a penalty.

So that's like a noiseless law so so does
it so you can always write a function of.

The defined on those and points as X.

times be.

So that's kind of obvious
a millimeter trying to put a cheer.

So a function F..

I think it's like this OK because

one two I if you say F.

zero is zero.

OK.

So these are my my victors my my B.

This is my B..

So and I some from one to on to so
I can write this some B.

K.

indicator OK a list and I.

Know.

So that's my next matrix.

OK so you get this X.

matrix these days I mix those B.'s So
you have a loss of problem so it fits in

the framework that we considered so
far OK then there's a little bit

miffed or a gossip going on that if you
lost so it won't work if the columns

of your design spaced highly correlated
Now let's look at this design

terrible high correlations between
the columns right correlation meaning in

a product if you take the inner
product of those two divide by N.

it's almost one and so
do huge correlations or

in the products this is a simple
still shows the loss of still works.

With I'm going to calculate the gold but
to build at the constant.

So I have a function and
say it's just piecewise constant.

Well it's always it's just
defined on a few points so

it's like this maybe like this and
then like this and then let's see.

This is to the point where
the first jump to securing and

here is the second jump and so on.

And.

So did the distance between
between jumpstarted the D.J..

Then the Compared to be that
the constant is equal to this so

there's an exact expression for
that quantity here.

Which depends on
the distance between jumps.

So this is an example where you
can really calculate that thing.

OK And these are to the so

it's a minimizer over some beast and

the beast except murder
minimum is a thing or dose B.

Well it's the inverse You see
to infer Cis of the jumps.

In first of the just as between
jumps now if you then if you have to

mint condition in a bet the mint
condition now says that the jumps.

Have to have a certain the size
this didn't have a certain size

OK so if you this is between jumps
It's like the same order as N.

the size is the moralists noise level
with Otherwise it has to be large.

And then you get this for this particular
case you get you know it's the.

Still up the squared times
to compensate beauty

as Ciro defied by the compatibility
constant foods like these.

And if you look out skills so
if that this is between jumps is

the same old a time and
if they're as Ciro or

as serial plus one So
you've got you can calculate the first.

The value where the function starts is
the free parameter so let's put an X.

if parameter so you have a serial
jumps plussed where it starts so

it's a serial this one free parameters in
the sense that it's a serial close one.

If equal jumps then the maximal
one over the jumps of maximal

value is done this year Opus one to
five by and you do the calculations and

you see that the prediction error
is order the number of jumps

squared times loken over and where
Logan over and is just up the squared.

But anyway if you if you have only a one
or two jumps you get a parametric rate

OK parametric rate in the sense of one
over and the locked of course because

cos you are you don't know where to
jump start and you can extend this to

trees if you have a total variation
penalty on trees you just for

instance for insist just cut up the trees
into Paul figure out and you do it for

each boss and you get similar results so
you can do this for more general graphs

OK not just to finish up a little bit
quickly you can do the noisy case like.

Like before but that's it gets a bit
technical I misstate it so it may be good

not to go to go into all the technical
details but you get something like.

Because of the noise you get in
the Complete the BT constant some weight.

So they're here and that see

just to say how it goes
though it's a result of

of papers are not young and
know how to beauty and

your harness Laden are very clever it
says OK all you have to do is kill

it somehow Why do you use a penalty in the
noisy case it is because there's no choice

it's almost to spend all to you have to
get rid of the noise kill the noise and

what the amount of noise it has to appeal
is less if you have higher correlations

because it's what you do you have your
active Ferial sure non active variables.

And then you just project those known
active variables on the active forms and

what's left over that's the part of
the noise that you have to kill the inner

product of that Barclays' the noise so
if if there's much correlation and

you project and then see what's left over
then there's hardly anything left over so

there's hardly any noise to kill so

here is this all defined this is to
project all this is just a usual thing if

you do square scene no this is
the yeah the protection of the non

active faerie was in the active ferryboats
and this is what's left is left over.

And this is then the go variance
matrix of what is left over

and then you look at the diagonal make
elements of this matrix and more or less

what you have to kill the noise you need
to your tuning parameter to be larger so

that you put this this quantity without
the square in the weights here.

OK so if there are more correlation
this will be very small and so.

You put one minus that quantity into it so
it will be almost one book

should be somewhere that's what I'm saying
is you only need that part of the kill

the part of the noise that is left over
after projecting on the active sit and so

if you have high correlations
not much to kill.

OK And
then the weights stay close to unity and

then you're going almost
a constant one here.

So let me just didn't
give the result is so

the upper bound is from the spacer or
it's also in other papers but

I think the real result is in
this one young a beauty and

later it was fairly nice day you see
the Compared to be the sequence and

they actually have it and

instead of a one you have one minus
something small here depending on those

projections Yes we see you can lump
the square with a serial defined

by accountability constant like like
being there but now for the noisy case.

And a lower bound is similar to
the bit I'll skip all the conditions

the lower bound dislike this.

These are small order terms lumped at
times the sparsity and then the competent

be the constant and with instead of
a minus you have a plus now and then you

just have to hope that the two are close
to each other it depends on the design OK.

Just to finish if you apply that to
the total variation penalty in the noisy

cases for now it's gets really
interesting because that's what

we do we have a noisy observation
of a function with a should

we do that though to variation penalty
we calculate those weights this

is really hard work I must say that it's
done in the paper by dollar Leon and all.

So and then you get the following result.

So the upper bound was known.

And for the lower bounds you
applied previous results and

for the special case you get this and

so you get the lower bound and the upper
bound they differ only in this term

as you get really quite tight bounds for
the total variation penalty and so

it depends this is in general
not a dominating therm.

So what do you see yeah so if you have
only a finite the many jumps a three

then this will be the dominating third and
so you get a parametric rate that slipped.

Parametric rate one over a square root
of IN THE me put it down again look.

Then.

In order.

Here.

So this is the one over square with
a friend it's a paramedic rate

then there's a look B.

term which is peace now N.

So here's a look enter and
then there's an additional local term so

you get the parametric rate.

A lot are not the square root
locked on but a lot better.

And if the number of jumps is
growing then this will probably not

depend on our flat for our church but
then this can be the dominating term.

And I'm I don't know whether this is
to the tightest possible result but

at least it's a two locked rooms
you know see how tight it is.

Of course you can this can also be
extended to the more general graphs

this is work with with the.

Functions Quarterly that
I've just bits on archive.

So let me give a summary so for
random the size you need these I

can value conditions but
then the bias dominates the variance and

for the bias you have except
expressions for the prediction error

under these bit I mean conditions and for
fixed the science there are upper and

lower bounds but then there may be
you get because of the constant here

and you know it's it's
not a priori clear how

big the gap is depends really
on that the sign OK Now name me.

This is all nonsense you go
through look at that so I have

to acknowledge some people I talk to from
conflicts up to my station in the local

fix up to my station to help me out
thank you very much for your attention.

Yes.

In I don't think so.

Ever infinite variance or.

OK.

Let me think you get a similar bounce but
not with exponential probabilities.

I guess that would be my guess depend so
if the design is bounded at least.

So bounded exis.

Then then I could do similar things but
then the probability is just

not going exponentially fast to zero the.

These.

Are.

Exactly.

The.

Nodes it's OK Just you left your noice.

Which is affected and you have another
factor which is fixed you just need

some banks of high probability for
quantities like this and

if there's a second moments you
know you don't with many of them so

if their second moments you can use
got some kind of what is called again.

Forgot it's called good if you
know me from bounce Also if only

if you only have second moments.

So it should work.

Thank.

You for the old saying.

What will the Obama look.

Like them so far I got no one
no source there you go see cool.