The important equation is the following:

Here I'll explain the parts of the formula using the following motivating example:

Suppose our true distribution is a normal distribution with unit variance, centered at 0, and we estimate its mean from data using method of maximum likelihood. This falls into domain of Empirical Risk Minimization because it's equivalent to finding unit variance normal distribution which minimizes empirical log-loss. If this method is consistent, then our maximum likelihood estimate of the mean will be correct in the limit

In order for the theorem to apply, we need to do make sure that we can put a bound on the loss incurred by functions in our hypothesis space, so here, make an additional restriction that our fitted normals must have mean between -100 and 100.

Here's what parts of the equation stand for

1.

l is the size of our training set

2.

a is the parameter that indexes our probability distributions, in this case, it's mean of our fitted Normal. Lambda is it's domain, in this case,

3.

This is the actual risk incurred by using estimate a instead of true estimate (0). In our case the loss is log loss, so the risk is

In other words it is the expected value of the negative logarithm of the probability assigned to observations when observations come from Normal centered at 0, but we think they come from Normal centered on a

4.

This is the risk incurred by using estimate a, estimated on a finite sample drawn from the true density. In our case it is the following

Because the sample is a random variable, empirical risk is also a random variable. We can use the method of transformations to find it's distribution. Here are distributions for empirical risk estimated from training sets of size one for different values of a.

You can see that the modes of the distributions correspond to the true risk (a^2), but the means don't.

5.

Supremum is defined as the least upper bound, and behaves lot like maximum. It is more general than the maximum however. Consider the sequence 0.9, 0.99, 0.999, ....

This sequence does not have a maximum but has a supremum.

6.

This is the simplified middle part of the equation. It represents the probability of our empirical risk underestimating true risk by more than epsilon, for a given value of a. This would be used in place of the supremum if we cared about pointwise as opposed to uniform, convergence. Intuitively, we expect this probability to get smaller for any fixed epsilon as the sample size l gets larger. Here are a few densities of risk distribution for a=4 and different values of l

7.

This is the random variable that represents the largest deviation of the empirical risk from corresponding true risk, over all models a. It's like an order statistic, except the samples are not IID, and there are infinitely many of them.

Questions:

- How can one find the distribution of \sup(R(a)-R_emp(a) in the setting above?

## 15 comments:

Hi Yaroslav.

I'm learning Bayesian nets in CS 430 -- it's fun. I just wanted to say "hi" on your blog. Hope you're having fun in school.

P.S.: If you haven't seen it yet, the Banach fixed point theorem gives a nice test for convergence of iterated functions.

Some of people do the post aicm now and they do it before usmle. It is really surprising that they choose this first and it could be tough at beginning but they can easily get it later. nice article & it will give you some amazing idea and very effective for the papers writing.

While I was going to learn some theory it really gives me pain as ther ehas hardship but here are some tips which help me out. website you'll be find some helpful information about the academic writing service.

There has nothing special you can do in the theory side if you do not study and try to practice it in your own way at home. look here for the students that is very helpful for the writing services.

My best friend send me the link and told me that you go to the page and visit there. Because this page share their post for the people like you. That's why I'm here. SO good really. Thanks to my friend for this link.

I am weak in The Key Theorem of the Learning Theory but I am very well with graf. I see you posted both of information here. its impresed me. I will check more info from your website in future. thanks

Excellent machine learning blog,thanks for sharing...

Seo Internship in Bangalore

Smo Internship in Bangalore

Digital Marketing Internship Program in Bangalore

Excellent machine learning blog,thanks for sharing...

Seo Internship in Bangalore

Smo Internship in Bangalore

Digital Marketing Internship Program in Bangalore

All The Interested Candidates Read The Notification of The Following Vacancy. Read and Check Are You Eligible For This Vacancy and Read The How to Apply The Online Form of This Job. Online Quran teacher.

Great Article

IEEE final year projects on machine learning

JavaScript Training in Chennai

Final Year Project Centers in Chennai

JavaScript Training in Chennai

Nice one, I guess we just need to stop for a brief moment and read through. I dont read that much but I really had some good time grabbing some knowledge. Still I am out to speak and propose you exotic stuffs like

exotic carts official

buy exotic carts

buy exotic carts online.

exotic carts review

exotic carts website

gorilla glue exotic carts.

exotic carts fake

kingpen official

kingpen review.

mario carts official

mario carts online

exotic carts website

exoticcartsofficial

exotic carts for sale

exotic carts thc

exotic cartridges

exotic carts flavors

@exoticcartsofficial

real exotic carts

exotic carts vape cartridges

exotic cart

exotic carts vape pen

mario kart cartridge

king pen battery

exoticcarts

exotic carts official website

supreme cartridges

mario carts cartridges

exotic carts review

710 kingpen gelato

710 king pen battery

supreme cartridge

supreme brand oil cartridge

what are exotic carts

what are pre roll

what are dabwoods

Some of the most successful ABS-CBN programs from the 1990s up to the present

bear his imprint — from “Showbiz Lingo”, “The Buzz”, lambingan - Watch your favorite Pinoy Tambayan, Pinoy Teleserye Replay, Pinoy

TV Series and Pinoy TV Shows online for free! “Today With Kris”, “Game Ka Na Ba?”, “Mula

Sa Puso”, “ASAP” and recent hits “May Bukas Pa”, “Tayong Dalawa”, “Walang Hanggan” and “Ang Probinsyano”.

gogo animeWatch anime online in English. You can watch free series and movies online and English subtitles.lottery sambad 2020is the Indian lottery system. The important and main thing which attracts the people to invest in this lottery is the number of prizes and size of the lottery. There are many other lotteries in India but sambad is a very popular lottery among the Indian people.In each dialogue box, the movies chronologically segregated in descending order.

This site is very organized and very possible. The film can be searched alphabetically.

From 0-9 and the alphabets individually can be searched.Hollywood, Bollywood, Malayalam, Tamil

and Telugu movies. Once you open the website, you will be able to see a range of options.123Movies,

GoMovies, GoStream, MeMovies or 123movieshub,movierulz,hindi movies,Bollywood | Hollywood | Songs | Punjabi

| Hindi Dubbed | Songs | Hindi Webseries Season | Netfilix | ULLU App

In a separate dialogue box, we can search by the plethora of the genres provided to us. Action, animation, adventure,

biography, comedy, crime, documentary, drama, family, fantasy, film-noir, game show, history, horror, music, musical, mystery,

news, reality show, romance, sci-fi, sports, talk- show, thriller, war and western and various genres can be found here.

Any web site offered u . s . by means of vital info for work relating to. ดูการ์ตูน

Post a Comment