Relation between solution of linear regression and Lasso regression

Summary

The script delves into the concept of regularization in linear regression, particularly exploring the idea of encouraging sparsity in solutions. It contrasts L2 norm regularization with L1 norm regularization, explaining how L1 can lead to solutions with more zero values, effectively selecting important features. The discussion introduces LASSO, or Least Absolute Shrinkage and Selection Operator, as a method for achieving sparse solutions by penalizing the sum of absolute values of coefficients, making it a popular choice for feature selection in high-dimensional spaces.

Takeaways

• 🔍 The script discusses the geometric insight into regularization in linear regression and how it affects the search region for the optimal weights (w).
• 🧩 It suggests a method to encourage sparsity in the solution, meaning having more coefficients (w's) exactly equal to zero, by changing the regularization approach.
• 📏 The script introduces the concept of L1 norm as an alternative to L2 norm for regularization, defining L1 norm as the sum of the absolute values of the vector's components.
• 🔄 L1 regularization is presented as a method to minimize the loss function plus a regularization term involving the L1 norm of w, aiming to promote sparsity in the solution.
• 📐 The script explains that L1 regularization can be formulated as a constrained optimization problem, with the constraint being the L1 norm of w.
• 📊 The geometric representation of L1 constraints is a polyhedral shape, unlike the elliptical contours of L2 regularization, which may lead to hitting a point where some features are zero.
• 🎯 The hope with L1 regularization is that the optimization process will more likely result in a sparse solution where some features have exactly zero weight.
• 🌐 The script mentions that in high-dimensional spaces, L1 regularization tends to yield sparser solutions compared to L2 regularization.
• 🏷️ LASSO (Least Absolute Shrinkage and Selection Operator) is introduced as the name for linear regression with L1 regularization, emphasizing its role in feature selection.
• 🔑 LASSO is described as a method to minimize the loss with an L1 penalty, aiming to shrink the length of w and select important features by pushing irrelevant ones to zero.
• 📚 The script acknowledges that while intuitive arguments are made for the sparsity-promoting properties of L1 regularization, formal proofs are part of more advanced courses and are not covered in the script.

Q & A

What is the main idea discussed in the script regarding regularization in linear regression?

The script discusses the use of geometric insights to understand how regularization works in linear regression and suggests using L1 norm instead of L2 norm to encourage sparsity in the solution, which means having more coefficients set to exactly zero.

What does the term 'sparsity' refer to in the context of linear regression?

Sparsity in linear regression refers to a solution where many of the coefficients (w's) are exactly zero, which means that those features do not contribute to the model.

What is the difference between L1 and L2 norm in terms of regularization?

L1 norm is the sum of the absolute values of the components of a vector, while L2 norm is the sum of the squares of the components. L1 regularization tends to produce sparse solutions, whereas L2 regularization tends to shrink coefficients towards zero but does not set them to zero.

What is the geometric interpretation of L1 and L2 regularization in terms of the search region for w?

L2 regularization corresponds to a circular search region (an ellipse in higher dimensions), while L1 regularization corresponds to a diamond-shaped or L-shaped region, which is more likely to intersect the contours at points where some features have zero weight.

Why might L1 regularization be preferred over L2 in certain scenarios?

L1 regularization might be preferred when there are many features, and it is expected that most of them are not useful or redundant. It can help in feature selection by pushing the coefficients of irrelevant features to zero.

What is the acronym LASSO stand for, and what does it represent in the context of linear regression?

LASSO stands for Least Absolute Shrinkage and Selection Operator. It represents a method of linear regression that uses L1 regularization to minimize the model's complexity and perform feature selection.

How does LASSO differ from ridge regression in terms of the solution it provides?

While ridge regression (L2 regularization) shrinks all coefficients towards zero but does not set any to zero, LASSO (L1 regularization) can set some coefficients exactly to zero, thus performing feature selection.

What is the 'shrinkage operator' mentioned in the script, and how does it relate to L1 and L2 regularization?

A shrinkage operator is a term used to describe the effect of regularization, which is to reduce the magnitude of the coefficients. Both L1 and L2 regularization act as shrinkage operators, but they do so in different ways, with L1 potentially leading to sparsity.

What is the intuition behind the script's argument that L1 regularization might lead to more sparse solutions than L2?

The intuition is based on the geometric shapes of the constraints imposed by L1 and L2 norms. L1's flat sides may cause the optimization to hit points where many coefficients are zero, whereas L2's circular constraint tends to shrink coefficients uniformly towards zero without setting them to zero.

Can the script's argument about the likelihood of L1 producing sparse solutions be proven mathematically?

While the script does not provide a proof, it suggests that in advanced courses, one might find mathematical arguments and proofs that support the claim that L1 regularization is more likely to produce sparse solutions compared to L2.

What is the practical implication of using L1 regularization in high-dimensional datasets?

In high-dimensional datasets, L1 regularization can be particularly useful for feature selection, as it is more likely to produce sparse solutions that only include the most relevant features, thus simplifying the model and potentially improving its performance.

Outlines

00:00

🔍 Exploring L1 Regularization for Sparse Solutions

This paragraph discusses the concept of using L1 regularization as an alternative to L2 for linear regression, aiming to encourage the model coefficients (w's) to be exactly zero, thus promoting sparsity in the solution. The L1 norm is defined as the sum of the absolute values of the vector components. The regularization term involves minimizing the loss function plus a penalty term that is the L1 norm of w, scaled by a lambda parameter. This approach is contrasted with L2 regularization, which results in a different constraint on the search space for w, potentially leading to solutions where features are shrunk but not necessarily eliminated. The discussion suggests that L1 regularization is more likely to yield sparse solutions, where some features are entirely discarded, due to the nature of the constraint it imposes on the model coefficients.

05:02

📉 The Geometry of L1 Regularization and Sparse Solutions

The second paragraph delves deeper into the geometric interpretation of L1 regularization and its tendency to produce sparse solutions. It explains that the L1 constraint forms a diamond-shaped region in the search space for the model coefficients, which is different from the circular constraint imposed by L2. The hope is that as the loss function increases, the solution will first encounter the boundary of this L1 region at a point where some coefficients are exactly zero, leading to a sparse solution where only a subset of features are selected as important. The paragraph also introduces the concept of LASSO (Least Absolute Shrinkage and Selection Operator), which is a specific application of L1 regularization used to perform feature selection by shrinking some coefficients to zero and keeping others.

10:04

🛠️ Utilizing LASSO for Feature Selection in Linear Regression

The final paragraph highlights the practical application of LASSO in linear regression problems, especially when dealing with a large number of features, many of which may be redundant or irrelevant. LASSO is presented as a tool that can effectively push the coefficients of less important features to zero, thereby simplifying the model and potentially improving its performance by focusing on the most relevant features. This approach is particularly useful in high-dimensional settings, where the goal is to achieve a more interpretable and robust model by reducing the dimensionality through feature selection.

Keywords

💡Geometric Insight

Geometric insight refers to the understanding of a concept through its geometric representation or visualization. In the context of the video, it is used to explain the workings of regularization in linear regression, particularly how it shapes the region from which the optimal weights (w's) are sought, affecting the contours of the loss function.

💡Regularization

Regularization is a technique in machine learning used to prevent overfitting by adding a penalty term to the loss function. The script discusses how this technique can be adapted to encourage the model weights to have zero values, leading to a sparser model. Regularization modifies the search region for the optimal solution, impacting the model's complexity.

💡L1 Norm

The L1 norm, also known as the Manhattan norm, is a measure of the size or length of a vector in space, defined as the sum of the absolute values of its components. The video explains that using the L1 norm for regularization can lead to a sparse solution, where some of the model's weights are exactly zero, unlike the L2 norm which tends to shrink weights but not set them to zero.

💡L2 Norm

The L2 norm, also referred to as the Euclidean norm, measures the straight-line distance from a point to the origin in Euclidean space. In the script, it is contrasted with the L1 norm, highlighting that L2 regularization tends to shrink all weights towards zero but does not set them to zero, resulting in a less sparse model compared to L1 regularization.

💡Sparse Solution

A sparse solution in the context of machine learning refers to a model where many of the weights are zero, effectively reducing the number of features used in the model. The video script motivates the use of L1 regularization as a means to achieve sparsity, which can be beneficial when dealing with many irrelevant or redundant features.

💡Constrained Optimization

Constrained optimization involves finding the best solution for an optimization problem subject to certain constraints. The video script describes how regularization with L1 or L2 norms can be framed as constrained optimization problems, where the constraints are related to the norms of the weight vector.

💡LASSO

LASSO stands for Least Absolute Shrinkage and Selection Operator. It is a type of regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the statistical model it produces. The script explains LASSO as an application of L1 regularization in linear regression, aiming to produce sparse models.

💡Shrinkage Operator

A shrinkage operator in the context of LASSO refers to the mechanism by which the magnitude of the model's coefficients is reduced, effectively shrinking the size of the weight vector. The script uses this term to describe how both L1 and L2 norms act as shrinkage operators, but L1 has the added benefit of promoting sparsity by pushing some coefficients to exactly zero.

💡Feature Selection

Feature selection is the process of identifying and choosing the most relevant features or variables from a dataset for use in model construction. The video script discusses how LASSO, through its L1 penalty, can be used for feature selection by driving irrelevant feature weights to zero, thus simplifying the model and potentially improving its performance.

💡Overfitting

Overfitting occurs when a model is too complex and captures noise or random fluctuations in the training data. This can lead to poor generalization to new data. The script implies that regularization techniques, such as L1 and L2, are used to prevent overfitting by controlling the complexity of the model.

Highlights

Geometric insight into how regularization works in linear regression.

Proposal to encourage exact 0 values in the weights (w's) for regularization.

Introduction of the concept of sparsity in solutions.

Explaining the L1 norm as an alternative to L2 norm for regularization.

Definition of L1 norm and its application in regularization.

L1 regularization as a method to achieve sparsity in feature selection.

Visual representation of L1 constraint region in comparison to L2.

Intuitive argument for why L1 regularization might lead to sparser solutions.

Explanation of how L1 regularization can result in a solution with fewer non-zero features.

Concept of LASSO and its abbreviation meaning.

LASSO as a method for feature selection and its significance.

Explanation of the 'shrinkage' aspect of LASSO in terms of parameter length.

LASSO's role in pushing irrelevant features to zero for feature selection.

LASSO's popularity in solving linear regression problems with many features.

Theoretical and practical implications of L1 regularization compared to L2.

Advanced courses might delve into proving the sparsity guarantee of L1 regularization.

Transcripts

 Now we have a geometric insight  

into how your regularization kind of works. Can  we use this insight to come up with perhaps a  

different way to regularize our linear regression?  So that we somehow encourage our w's to have  

exactly 0 values. So, in other words, can we  change this region from which we are searching for  

our w, which is what your regularizer essentially  does. So, it is putting a constraint on where you  

are searching for w. Can we somehow change this  so that it is more likely that these elliptical  

contours are going to hit at a point where some  of the features exactly becomes zero and not,  

it just become small? So, this is a motivation to make to  

kind of get a sparse solution, sparse means a  lot of 0s. And an alternate way to regularize  

would then be using, as we will see why this is a  better way, is using L1 norm instead of L2 norm,  

L2 squared norm. What does that mean?  So, what is an L1 norm? Well, L1 norm  

of a vector w is defined as sum over i equals 1 to  d, the absolute value of each of its components,  

we are just summing up the absolute values. Now, what does this mean? This means that if  

we are, now if we are regularizing using the  L1 norm, which would look something like this,  

regularization, L1 regularization would  mean that we are trying to minimize over w,  

the loss is the same, sum over i equals 1 to n w  transpose xi minus yi squared, now, plus lambda  

L1 of w, which is the sum of the absolute values. Now, our equivalence should tell us that this is  

equivalent to minimizing w in Rd, the loss as  usual, but then now your, for a given choice  

of lambda, it is going to be some other theta,  which will impose a constraint on the L1 norm.  

It may not be the same theta as the  L2, but then we are just going to  

use abuse notation and say it is also theta. So, some theta, so it does not matter what the  

value is, the more important thing is that you can  write the same problem with the L1 regularization  

also as a constrained optimization problem,  where now the constraint is on the L1 norm of w.  

Now, what does it mean to say? We  have an L1 constraint on w. So,  

how does the picture now look like? Well, first question is, where are the L1  

constraint w's present? So again, we go back to  the picture. So, earlier, we had this guy. So,  

which was our L2 constraint, this was our  L2 constraint. So, just norm w squared  

less than or equal to theta. Now, we are not  searching here, we are searching elsewhere. We  

are searching in L1 norm of w less than or equal  to some other theta. But where are those guys?  

Well, if you think about where those guys are,  well, those are exactly here in this region.  

So, this is norm L1 less than or equal  to theta. Well, let me shade this.  

This is the region we are searching for our w.  

Now, why is this helpful? Well, again, this is an  intuitive argument why this may be helpful. Sorry.  

Let us say we again had our  w hat ML somewhere here.  

And we had our elliptical  contours around w hat ML.  

Now, what might happen is as you increase the  loss, so that is still happening this way only.  

So, as the loss increases in an elliptical  fashion, the hope is that you will first hit  

somewhere here. So, I am not drawing it that well,  but then the hope is that you will hit it at this  

point, more likely, it is not absolutely necessary  you are going to hit here. But if you hit here,  

as opposed to hitting here, so if you consider  this would be our w hat ridge, because that is  

the first place where I hit the circle, whereas,  because there is a bulge you tend to hit before.  

So, before you hit this flat surface, so you move  further, when you are doing L1 penalisation. And  

you might hit somewhere here. So, this is w  hat. Well, I will give the name, so L. So,  

this is w hat L, and we will say what L is,  this is equivalent to the L1 regularization,  

regularized solution. But what is the use of  hitting it at this point? Well, at this point  

you have only one feature, which has a  positive value, the other feature has 0 value.  

So, if you see there are only four,  because there are only two features  

spare solution will just pick one feature among  these two features. So, you either pick both the  

features or you cannot, you cannot not pick both  the features, which means you cannot just give 0  

values to both the w's so that is meaningless.  But here it is sparsity would mean that you are  

just picking one feature and the only places where  you can pick one features are these four points.  

So, these are the four points where one of the  coordinate value is 0. So, which means that one  

feature has weightage 0. The other feature only is  important. So, now, in two-dimension one can argue  

that why is w hat ml here, why are the elliptical  contours like this can it be placed differently  

such that you may not get such a sparse solution,  I agree that can happen. But in high dimension,  

what typically people observed is that when you do  an L1 regularizer, you are more likely to hit more  

sparse solution than an L2 regularizer. So, the intuitive argument is that L2  

has this bulge in terms of where this  feature space where you are searching,  

but then L1 is flat and so you kind of move beyond  the bulge and then hit a flat point where a lot of  

many competence might be 0. This is an intuitive  argument. Of course, one can prove a few things  

about this, we will not do that, but then in  an advanced course, you would try to prove  

when your L1 is guaranteed to give you sparse  solution and so on, we are not doing that.  

For now, this would suffice to say that instead of  the L2 regularizer if you used an L1 penalty or a  

regularizer then you perhaps will get more sparse  solution. So, this way of doing this doing linear  

regression is what is called as L1 regularization,  this name for this is what is called as LASSO.  

LASSO is an abbreviation for as an  acronym for Least Absolute Shrinkage  

and Selection Operator.  

So, what does this mean? It just means that, I mean each  

word here has a meaning that we already kind of  have seen. It is least because we are minimizing  

some loss. Absolute is because we use the L1  penalty, which is just a sum of absolute values.  

It is called shrinkage because the length  of w is shrunk. So, we are only searching  

in a space where w has smaller length and so  this is a shrinkage operator. And even L2 is  

the shrinkage operator for that matter. So, anything that minimizes the length,  

I mean such space by minimizing the length of the  parameter to search for is a shrinkage problem.  

So, this is also a shrinkage operator, but then  it shrinks using the absolute value that is the  

difference Selection because the hope is  that you are not just shrinking to make  

w value smaller but you eventually want  this to select the important features.  

So, you want to push a lot of w's to 0, components  to 0, exactly 0 such that the remaining features  

which get non-zero values are the ones that are  important for minimizing the loss. So, we can  

select only those features and then leave out  the rest. So, this is also a selection problem.  

Operator is just fancy word to  say that it is a regularizer.  

So, this is what is called as the  LASSO penalty or LASSO problem.  

And this is also very popularly used to  solve the least linear regression problem,  

especially when you have like a lot of  features, when you hope that most of  

these features are useless or redundant features  then LASSO would kind of push them to exactly 0.