Characteristics of Lasso regression

Summary

The video script discusses the advantages of Ridge regression over Lasso, despite Lasso's tendency to produce sparse solutions. It highlights the lack of a closed-form solution for Lasso due to its non-differentiability at zero, necessitating iterative methods like subgradient descent. The script also touches on special techniques for solving Lasso, such as the Iteratively Reweighted Least Squares (IRLS), and concludes with an overview of linear regression, including least squares, stochastic gradient descent, and various regularizers like Ridge and Lasso, emphasizing their applications in machine learning.

Takeaways

• 📚 The script discusses the comparison between Lasso and Ridge regression, highlighting the benefits and drawbacks of each method.
• 🔍 Lasso does not have a closed-form solution due to its non-differentiability at zero, unlike Ridge regression which has a straightforward closed-form solution.
• 🛠 To solve Lasso, subgradient methods are used because of the non-differentiability issue; these methods work even when the function is not differentiable at all points.
• 📉 The concept of subgradients is introduced as a way to approximate the direction of steepest descent for non-differentiable points in optimization problems.
• 📌 The script provides an intuitive explanation of subgradients, showing how they can be used to approximate the gradient in non-differentiable regions.
• 🔢 The absolute value function's subgradients are demonstrated, explaining how they can take any value between -1 and 1, representing different slopes that lower bound the function.
• 🔄 The definition of a subgradient is given, emphasizing its role in linearizing a function at a point and ensuring the function's value is always above this linearization.
• 🔧 Subgradient descent is presented as a useful algorithm for minimizing convex functions, even when they are not differentiable, by moving in the direction of the negative subgradient.
• 🔑 The relevance of subgradients to Lasso is highlighted, as Lasso is a convex optimization problem that can benefit from subgradient descent methods.
• 🛑 The script concludes that while Lasso provides sparse solutions, it lacks a closed-form solution and requires optimization techniques like subgradient descent or specialized methods like Iteratively Reweighted Least Squares (IRLS).
• 🚀 The summary of the course content on regression is provided, covering least squares, Ridge regression, Lasso, and the use of various regularizers in machine learning models.

Q & A

Why might someone choose Ridge regression over Lasso despite Lasso's ability to push some coefficients to zero?

Ridge regression has a closed-form solution, which makes it computationally simpler and faster compared to Lasso, which requires iterative methods like subgradient descent due to its non-differentiability at zero.

What is a closed-form solution and why is it beneficial?

A closed-form solution is an exact analytical expression that can be used to solve an equation or optimization problem. It is beneficial because it allows for direct computation of the solution without iterative methods, which can be more efficient and faster.

Why is Lasso not differentiable at zero and what are the implications for optimization?

Lasso is not differentiable at zero due to the L1 penalty term, which is an absolute value function that is not smooth at zero. This non-differentiability means that traditional gradient-based optimization methods cannot be directly applied to Lasso, necessitating the use of subgradient methods.

What are subgradient methods and how do they differ from gradient methods?

Subgradient methods are optimization techniques used for problems where the objective function is not differentiable at all points. They use subgradients, which are generalizations of gradients, to find a direction of descent even in non-differentiable regions. In contrast, gradient methods rely on the derivative of the function, which must exist at all points of interest.

Can you provide an example of a subgradient?

A subgradient for a piecewise linear function at a non-differentiable point can be any line that lies below the function and touches it at that point. For instance, in a function with a 'V' shape, multiple lines can serve as subgradients at the vertex, each representing a different direction in which the function is lower-bounded.

What is the definition of a subgradient in the context of convex optimization?

A subgradient of a function f at a point x is a vector g such that for all z, the function value f(z) is greater than or equal to f(x) + g^T (z - x). This means that the function lies above the linear approximation defined by the subgradient at x.

Why are subgradients useful in optimization, especially for non-differentiable functions?

Subgradients are useful because they allow for the optimization of convex functions that may not be differentiable. By moving in the direction of the negative subgradient, one can still converge to the minimum of the function, provided it is convex.

What is the relationship between the L1 penalty and subgradients?

The L1 penalty, which is the absolute value of a variable, is not differentiable at zero but has subgradients. At points other than zero, the subgradient is the sign of the variable, and at zero, any value between -1 and 1 can be a subgradient.

What are some alternative methods to solve the Lasso problem besides subgradient descent?

Besides subgradient descent, other methods include the Iteratively Reweighted Least Squares (IRLS) method, which leverages the structure of the Lasso problem by solving a series of weighted least squares problems.

What is the significance of the Lasso problem being a convex optimization problem?

The convexity of the Lasso problem ensures that any local minimum is also a global minimum. This property allows optimization algorithms like subgradient descent to find the optimal solution reliably, even though the problem may not have a closed-form solution.

Can you summarize the main differences between Ridge and Lasso regression in terms of their solutions and properties?

Ridge regression uses an L2 penalty and has a closed-form solution, making it computationally efficient. It shrinks coefficients towards zero but does not set them exactly to zero. Lasso regression, on the other hand, uses an L1 penalty and does not have a closed-form solution. It can result in sparse solutions with some coefficients exactly at zero, but requires iterative optimization methods.

Outlines

00:00

🔍 Lasso vs Ridge Regression and Subgradient Methods

This paragraph discusses the comparison between Lasso and Ridge regression, highlighting that Lasso does not have a closed-form solution due to its non-differentiability at zero caused by the L1 penalty. It explains the use of subgradient methods to solve such optimization problems where traditional gradient descent isn't applicable. The explanation includes an intuitive description of subgradients, using a piecewise linear function as an example to demonstrate how subgradients can provide a direction for optimization even when the gradient isn't defined. The paragraph concludes by emphasizing the iterative nature of solving Lasso regression problems.

05:02

📉 Understanding Subgradients in Optimization

The second paragraph delves deeper into the concept of subgradients, providing a formal definition and explaining their role in optimization, particularly for convex functions. It illustrates how subgradients can be used in place of gradients for optimization when the function isn't differentiable. The explanation includes a geometric interpretation of subgradients with respect to a one-dimensional absolute value function, showing multiple subgradients at points of non-differentiability. The paragraph also discusses the convergence properties of subgradient descent, a method applicable to convex functions, and its relevance to solving the Lasso problem, which is a convex optimization task.

10:05

🛠 Specialized Methods for Solving Lasso Regression

This paragraph summarizes the challenges and methods for solving the Lasso regression problem, emphasizing that there is no closed-form solution due to its L1 penalty. It mentions the iterative re-weighted least squares (IRLS) method as a specialized approach to tackle Lasso problems, leveraging the solvability of least squares as a 'black box'. The paragraph also touches on the possibility of using general optimization techniques like subgradient descent for Lasso, given its convexity, and acknowledges the existence of other specialized methods without going into detail. It concludes by contrasting the ease of solving Ridge regression with the more complex nature of Lasso.

15:05

🌟 Conclusion on Regression and Transition to Classification

The final paragraph wraps up the discussion on regression, providing a high-level summary of the concepts covered in the course, including least squares, stochastic gradient descent, maximum likelihood estimation, Bayesian linear regression, Ridge regression, and Lasso. It also mentions the existence of various regularizers and the possibility of incorporating domain-specific constraints into regularization. The paragraph concludes by signaling the transition to the next part of the course, which will focus on classification within the realm of supervised learning.

Keywords

💡Lasso

Lasso, short for Least Absolute Shrinkage and Selection Operator, is a type of linear regression that includes an L1 penalty term. It is known for its ability to produce sparse models, where some coefficients can be exactly zero, which helps in feature selection. In the video, Lasso is compared with Ridge regression, and its non-differentiability at zero is discussed, which makes it challenging to solve with standard gradient descent methods.

💡Ridge Regression

Ridge Regression is a linear regression technique that includes an L2 penalty term. It is used to analyze multiple regression data that has multicollinearity. Unlike Lasso, Ridge Regression does not produce sparse solutions; instead, it shrinks the coefficients towards zero but does not set them to zero. The script mentions that Ridge Regression has a closed-form solution, which is a significant advantage over Lasso.

💡Closed-form Solution

A closed-form solution refers to a mathematical expression that can be used to solve an equation exactly, without the need for iterative methods. In the context of the video, it is mentioned that Ridge Regression has a closed-form solution, while Lasso does not, due to its non-differentiable penalty term at zero.

💡Subgradient Methods

Subgradient methods are optimization techniques used for problems where the objective function is not differentiable at certain points. The video explains that since the Lasso penalty term is not differentiable at zero, subgradient methods are used to solve the optimization problem. These methods use subgradients, which are generalizations of gradients, to find the minimum of a function.

💡Gradient Descent

Gradient Descent is a first-order iterative optimization algorithm used to find the minimum of a function. In the video, it is mentioned in contrast to subgradient methods, as it cannot be used for Lasso due to its non-differentiability. However, it is applicable to Ridge Regression, which has a closed-form solution.

💡Convex Function

A convex function is a function whose second derivative is non-negative for all points in its domain. In the video, it is noted that the Lasso problem is a convex optimization problem because the sum of a quadratic loss function and an L1 norm (which is convex) results in a convex function. This property is important because it guarantees that any local minimum is also a global minimum.

💡L1 Norm

The L1 norm, also known as the Manhattan norm, is a measure of the length of a vector. It is used in Lasso regression as a penalty term to encourage sparsity in the solution. The video discusses how the L1 norm contributes to the non-differentiability of the Lasso penalty term at zero.

💡Iterative Re-weighted Least Squares (IRLS)

IRLS is a method for solving regression problems with non-differentiable penalties, such as Lasso. The video mentions IRLS as a specialized method for solving Lasso problems by iteratively solving weighted least squares problems. This method takes advantage of the structure of the Lasso problem to find the optimal solution.

💡Sparsity

Sparsity in the context of regression models refers to the property of having many coefficients set to zero. This is a desirable property in feature selection, as it simplifies the model and can improve interpretability. The video discusses how Lasso, unlike Ridge, can produce sparse solutions by driving some coefficients exactly to zero.

💡Regularization

Regularization is a technique used to prevent overfitting in machine learning models by adding a penalty term to the loss function. In the video, both Ridge and Lasso regression are discussed as examples of regularized linear regression, with Ridge using an L2 penalty and Lasso using an L1 penalty.

Highlights

Lasso does not have a closed form solution unlike Ridge regression, which complicates solving the optimization problem.

Subgradient methods are used to solve Lasso due to its non-differentiability at zero.

A subgradient is a direction where the function is completely lower bounded, used when the gradient is not available.

Subgradients can be thought of as slopes in one-dimensional cases, providing a way to approximate non-differentiable points.

The L1 penalty in Lasso has subgradients because it is not differentiable at zero, with slopes between -1 and 1.

Subgradient descent is an algorithm that can be used for convex functions even if they are not differentiable, converging to the minimum.

The Lasso problem is convex, allowing the use of subgradient descent methods for optimization.

Iterative Re-weighted Least Squares (IRLS) is a specialized method for solving the Lasso problem.

The Lasso problem combines a linear regression with an L1 penalty, promoting sparsity in the solution.

There are no closed-form solutions for Lasso, necessitating the use of optimization techniques or special purpose methods.

Ridge regression provides a closed-form solution and is easier to solve compared to Lasso.

Linear regression with squared loss is equivalent to the maximum likelihood estimator when given a probabilistic twist.

Bayesian linear regression introduces a prior distribution for the weights, leading to Ridge regression as a regularized version.

L1 regularization (Lasso) is preferred for promoting sparsity in the weight vector compared to L2 regularization (Ridge).

Elastic net penalty is a mixed regularizer combining Lasso and Ridge, catering to specific domain constraints.

Domain-specific constraints can be incorporated into regularizers to guide the optimization process towards expected solutions.

The course concludes with an overview of linear regression techniques and introduces the next part focusing on classification.

Transcripts

foreign

[Music]

so now the question is well why don't we

just do lasso right so if it's if it is

giving us

benefits in terms of uh you know the the

pushing the values to exactly zero why

would anyone even want to do rich when

lasso is easy where there are some

advantages of doing Ridge still than

lasso

foreign

I mean let me point out a few things one

thing is that one points some points um

Point number one is that lasso

does not have

a closed form solution

remember our Rich regression had a

closed form solution ah Bridge closed

from solution was x x transpose plus

Lambda I inverse X transpose y right so

it was a closed form solution whereas

lasso you cannot say that well I cannot

take the derivative set it to 0

simply because the problem itself is not

you know exactly differentiable at zero

right so your your penalty is not a

differentiable function so you cannot

take the gradient set it to 0 and solve

for it

um so how do we solve this problem

well you can use what are called as

subgradient methods

are usually used

to solve lesson

remember our gradient methods were just

following the negative gradient

direction now

for problems where differentiation is an

issue right so where you cannot take the

gradient or at all points you can still

solve the problem using an iterative

method If instead of the gradient

something called as a sub gradient

exists a subgradient is equivalent to a

gradient if a gradient is present at a

point if not it is it is a direction

where the function is completely lower

bounded in that direction right so um

so I'll just give you some examples as

to what subgradients are we won't get

into too much detail about sub gradients

but just to give give you some sense of

how subgradients look like

let's say you have a point you have some

function like this

right so this is uh this is a piecewise

linear function right so you you see

there are different pieces of this

function at different points um

now at this point

right so at this point of the input um

let us say this is X now there are two

pieces which are intersecting at this

point and the problem is there is no

it's not differentiable at this point so

there is no gradient that you can find

for this point but then a sub gradient

is like

you know some

line right so some way to approximate

this function at this point which is

completely lower bounding the function

right so if if the function is

completely lower bounded at this point

um for example this is one subgradient

maybe here is another subgradient as you

as you can see there are multiples of

gradients here for the same point

um if

if you can compute such sub gradients if

your function has such the gradients at

every point for example at this point

its a different point

the only subgradient that you will have

at this point is this itself

this line itself no other line can

completely lower bound this function at

every Point whereas at these meeting

points there might be multiple lines

right so these are called as sub

gradients

um the blue one is also called a

subgradient but then because it's

differentiable at this point it is also

called as a gradient

right so if there is only one

subgradient well typically that's the

gradient right so

um that's kind of telling you how the

function is slope you can think of it as

a slope right so in one dimensional case

um so why is it this relevant for us

because the L2 penalty has ah is of is

like this right so it's an absolute

value function absolute value function

looks like this

right so it it looks like this ah this

is X this is absolute value of x and

then you can think of it's like two

pieces of linear functions type I mean

attached at this point so it is not

differentiable at this point uh but it

has sub gradients right so for example

this line is a sub gradient this line is

a subgradient well this line itself is a

sub gradient this line itself is a

subgradient right so all these lines

have slopes and all of these slopes are

subgradients right so it has multiple

sub gradients at this point in fact the

subgradient are sub gradient will just

be ah the slope for one dimensional

function at 0 is any value between minus

one and one minus 1 is this slope one is

this slope so anybody between minus one

and one any line is aligned with such a

slope will lower bound this function

absolute value completely and its called

as subgradient

right so um so let me quickly give you

the definition of sub gradient and then

we'll close this discussion so

subgradient uh I just intuitively said

what subgradient is but um you can think

of sub gradient as a vector

some Vector in D Dimension is a

subgradient

of some function f which is r d to R

at a point

X

in r t if

for every value of Z if the function's

value is greater than

the value at f of x plus G transpose Z

minus X which just means that if you if

you linearize this function at X right

so

um then the function

um

ah linear is using the sub gradient G

then the function is completely above

this linearization right so that is what

this means right so that is precisely

what this means so this um again so if

you want the example that we saw earlier

right so we had a function like this and

at this point it's not differentiable

but then at any Z right so I can let's

say I draw this function now I take some

Z the function itself takes the value F

of Z here so this is f of Z now if this

is the slope is G then this value is

just f of x plus G times Z minus X right

so that's what this value would be and

you can see that F of Z is above this

right so it's always about this now no

matter which set I pick wherever I pick

Z well the function is always above the

blue line and if you can find such a g

that satisfies this for all Z then

that's a sub gradient

now what's the use of sub gradient well

ah here is the reason why sub gradients

are useful ah y sub gradients

well if the function

f

to minimize

is a convex function

then

sub gradient descent is an algorithm

that one can use instead of gradient

descent

and this algorithm also converges

what does this mean this means that if

your function is nice then

instead of well even if it is not

differentiable if it is nice in the

sense that it's convex even if the

function is not differentiable you can

still Converse to the minimum of this

function by moving along the negative of

the sub gradient as opposed to the

gradient of course there are multiple

subgradients possible at a given point

where it is not differentiable it

doesn't really matter which subgradient

you pick you can pick one arbitrary

subgradient and move along the negative

of that subgradient direction and then

if the function is convex then one can

argue that a subgradient descent

algorithm will also converge to the

optimal solution why is this relevant

for us well it is relevant for us

because the lasso problem is a convex

problem it is the loss function is

quadratic the norm L one Norm is a

convex function so the sum of convex

functions is convex and so the lasso

problem itself is a convex optimization

problem and providing the minimum of a

convex optimization problem you can now

use something like a subgrade in the

center

um

by no means this is the only way to

solve this lasso problem there are other

specialized methods that one can use to

solve the lasso problem

um so we won't look at those methods but

I will just point out that there are

other methods

special purpose methods

because this is a special quadratic loss

plus plus

L one penalty now you can develop

special purpose methods to solve this

problem right so for lasso

uh one example is what is called as irls

um which is the iterative

re-weighted

least Quest method

again we won't really look at this

problem this method in this course

but

essentially the idea is that we can

solve least queries easily right so Lee

square is the original linear regression

problem we know how to solve in closed

form now you can use the use that as

some kind of a black box to solve the

lasso problem and now essentially this

is you know taking advantage of the

structure in the lasso problem that it

is a linear regression problem plus a

penalty and you can do special purpose

things like irls which can be used to

solve this problem that's one thing or

you can use a general purpose of

gradient descent algorithm by noting

that our function loss function plus

penalty the lasso objective is in fact

convex it may not be differentiable but

it's convex and so you can use subgrade

InDesign methods so all these are ok

just that we have to be aware that there

is no closed form solution to the lasso

problem and that's the main summary of

this this part of the you know

discussion there is no close form

solution to lasso so we cannot you know

hope that given the data and labels we

can simply really get a close form value

for w we have to do some work and that

is important to keep in mind though you

will get a sparse solution and all that

it is not as straightforward as solving

a ridge regression problem on the other

hand is gives you a nice close form

solution

you can do a stochastic gradient descent

also if you wish if you have large data

size n large D and so on but for me for

you know small values you can just use a

closed form solution small values of N

and D you can use the closed form

solution whereas in lasso there is no

closed form solution and so you have to

rely on you know either optimization

techniques like subgrade indesant or

special purpose methods like IRS

okay so so with this uh we we kind of

conclude our discussion about uh the

regression problem

I just wanted to summarize at a high

level whatever we have seen so far in

regression

um and then we'll move on to the next

part of the course in regression we

noted that uh you know you can solve a

least squares objective which is

required loss and that had a closed form

solution we looked at its geometric

interpretation we looked at its

computational

discussion we had a discussion and we

came up with the stochastic gradient

descent algorithm then we noted that

well if you give it a probabilistic

Twist then the linear regression with

squared loss is same as the maximum

likelihood estimator now because it's a

maximum likelihood estimator then we can

ask well is there a Bayesian counterpart

to this and that gave us the linear

regression plus regularized version

which is the rich regression problem

where you used an L2 regularizer which

was equivalent to a gaussian prior with

zero mean on the W itself and then we

noted that well the while the L2

regularizer is good in pushing w W's to

close to 0 it may not make it exactly

zero now to make it exactly zero then we

looked at the geometry of the problem

and said that well maybe an L1

regularizer is better and that gave us

the lasso problem the least absolute

shrinkage selection operator algorithm

which is the loss squared loss plus an

L1 regularizer and we also had a small

discussion about how to solve the lasso

problem involving sub gradients and

other techniques perhaps like the irls

now this is the overall summary of

linear regression I also wanted to point

out that you know people use all sorts

of regular razor there are mixed lasso

plus Rich regularizer called the elastic

net penalty and so on so there is a

whole you know um you know cottage

industry of regularizers where if you

think your W should satisfy some nice

constraint maybe W has some group

structure in the sense that you collect

features from lot of places but you know

that some features should either be

positive or sorry either be selected

together or not be selected together

these These are domain specific

constraints that might come up and then

you can convert them into you know

regular razors and then try to solve the

problem in a regularized fashion all

these are things that people have

studied um what you what we have looked

at in this course is some basic type of

regularizes which are very popular which

are most commonly used but then you also

should as machine learning practitioners

you also should know that in case

if your domain knowledge tomorrow in

your problem that you try to solve you

have some domain knowledge with say

something more about the structure of

the solution that you expect to see then

you can perhaps convert that into a more

meaningful regularizer than simply using

a ridge or a lasso penalty with this we

will complete our discussion about

regression in this course and we will

move on to the next part of the course

where we will talk about supervised

learning but from a classification point

of view thank you