Ridge regression

Summary

This script delves into the concept of ridge regression, a method that introduces regularization to linear regression to avoid overfitting by penalizing large weights. It explains the addition of a term scaled by lambda to the loss function, which encourages smaller weight values and helps in reducing redundancy among features. The script also touches on the Bayesian perspective, where lambda is connected to the prior distribution's variance, and discusses the balance between minimizing loss and avoiding redundancy in model weights.

Takeaways

• 📉 Linear Regression in Machine Learning is often defined as finding the best 'W' that minimizes the sum of squared differences between the predicted and actual values.
• 🔍 The script introduces a new estimator called 'Ridge Regression', which is a modification of the standard linear regression to include a regularization term.
• 🔑 The 'R' in Ridge Regression stands for 'regularization', a concept with roots in classical statistics, aiming to prevent overfitting by adding a penalty term to the loss function.
• 🎯 The regularization term is a scaled version of the norm of 'W' squared, controlled by a hyperparameter 'lambda', which needs to be determined through cross-validation.
• 🧩 The addition of the regularization term is akin to imposing a prior on 'W', suggesting a preference for solutions with smaller magnitudes to avoid redundancy in features.
• 🌐 The 'lambda' hyperparameter corresponds to the inverse of the variance in a Gaussian prior distribution over 'W', indicating the strength of the belief in feature redundancy.
• 📉 The script explains that a smaller variance (and thus a larger 'lambda') implies a stronger penalty for having large weights, discouraging the selection of redundant features.
• 🔎 The concept of redundancy is illustrated with an example where multiple linear combinations of features could explain the same label, and the goal is to select the simplest combination.
• 📍 The script suggests that pushing as many feature values towards zero as possible is a strategy to avoid redundancy, assuming most features are not necessary for the prediction.
• 🤔 It raises the question of whether Ridge Regression is the only way to achieve regularization and hints at the existence of alternative approaches and formulations.
• 🚀 The script concludes by suggesting that understanding Ridge Regression in a different geometric context could lead to better algorithms and formulations of the linear regression problem.

Q & A

What is linear regression in the context of the provided script?

Linear regression, as described in the script, is a method in machine learning where the goal is to find the best fitting line through the data points by minimizing the sum of the squared differences between the predicted and actual values (W transpose xi - yi) squared for all i from 1 to n.

What is the role of the estimator W hat M L in linear regression?

W hat M L represents the estimator that minimizes the sum of squared errors in linear regression, which is the optimal set of weights that best fits the data according to the least squares method.

What is the significance of the term 'ridge' in ridge regression?

The term 'ridge' in ridge regression comes from the concept of ridge estimation, which is a technique used to analyze multiple regression data that suffer from multicollinearity. It is a method that adds a degree of bias to the regression estimates in order to reduce the variance.

What does the term 'W hat R' represent in the script?

W hat R represents the estimator in ridge regression, where 'R' stands for ridge. It is the set of weights that minimizes the sum of the squared differences between the predicted and actual values, plus a regularization term (lambda times the norm of W squared).

What is the purpose of adding the regularization term in ridge regression?

The regularization term (lambda times the norm of W squared) is added to the loss function in ridge regression to penalize large weights and prevent overfitting. It encourages the model to prefer solutions with smaller values of W, reducing the model's complexity.

What is the Bayesian viewpoint of the regularization term in ridge regression?

From a Bayesian perspective, the regularization term can be seen as arising from a prior belief about the distribution of the weights (W). It is as if we are placing a prior on W that prefers smaller values, which is equivalent to saying we believe most features are redundant and should have small or zero weights.

What is the role of lambda in ridge regression?

Lambda is a hyperparameter in ridge regression that controls the amount of shrinkage applied to the coefficients. It determines the trade-off between fitting the underlying data and keeping the model weights small to avoid overfitting.

How does the prior assumption of zero mean for W in ridge regression influence the model?

The prior assumption of zero mean for W implies that we prefer weights to be as close to zero as possible, effectively reducing the influence of redundant features and promoting simpler models with fewer active features.

What is the concept of redundancy in the context of features in a regression model?

Redundancy in the context of features refers to the presence of multiple features that provide similar information about the target variable. This can lead to multicollinearity and make it difficult for the model to determine the individual contribution of each feature.

Why is it not always desirable to push W to zero in ridge regression?

While pushing W towards zero can help reduce redundancy and overfitting, setting W to zero entirely would result in a model that does not make any predictions, which is not useful. The goal is to find a balance where the model is simple but still captures the underlying patterns in the data.

What is the potential advantage of having some weights in W to be zero in a regression model?

Having some weights in W to be zero can be advantageous as it simplifies the model by eliminating the influence of redundant or irrelevant features, potentially leading to a model that is easier to interpret and generalizes better to new data.

How does the value of gamma squared relate to the lambda hyperparameter in ridge regression?

The lambda hyperparameter is inversely related to gamma squared, the variance term in the prior distribution of W. A smaller variance (stronger belief in redundancy) leads to a larger lambda, which in turn increases the penalty for larger weights and encourages more weights to be zero.

What are some alternative approaches to regularization besides ridge regression?

Besides ridge regression, other regularization techniques include Lasso regression, which adds an L1 penalty term, and Elastic Net, which combines L1 and L2 penalties. These methods can be more effective in certain scenarios and can lead to different types of solutions.

Outlines

00:00

📊 Introduction to Ridge Regression and Its Bayesian Perspective

This paragraph introduces the concept of ridge regression as an extension of linear regression. It explains that linear regression is about finding the best fit line (W) that minimizes the sum of squared differences between the predicted and actual values. The paragraph then introduces a new estimator, known as the ridge estimator, which includes an additional term to the loss function—a scaled version of the norm of W squared, controlled by a hyperparameter lambda. This term acts as a regularizer, penalizing larger values of W to avoid overfitting and to prefer solutions where the weights are as small as possible, reflecting a prior belief that the true weights are likely to be close to zero. The explanation connects this regularization to Bayesian statistics, where the lambda corresponds to the inverse of the variance in a Gaussian prior distribution over the weights, indicating a preference for solutions with smaller magnitudes in the weight vector.

05:05

🔍 The Role of Redundancy and Regularization in Feature Selection

The second paragraph delves into the implications of feature redundancy in the context of linear regression and the role of regularization in addressing it. It discusses how when dealing with a large number of features, many of which might be redundant, it's beneficial to select a set of weights that minimizes redundancy. The paragraph uses an example involving height and weight to illustrate how multiple combinations of features can explain a given label, but the goal is to choose the combination that has the smallest 'length' or norm, which corresponds to having the least redundancy. The explanation of lambda continues, indicating that a smaller variance in the prior (a stronger belief in redundancy) leads to a larger lambda, thus a higher penalty for large weights. This trade-off between minimizing loss and avoiding redundancy is central to ridge regression, and the paragraph concludes by hinting at alternative approaches to regularization and the potential for a geometric understanding of ridge regression that could lead to different formulations or algorithms.

Keywords

💡Linear Regression

Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In the video's context, it is the fundamental concept being built upon with the introduction of the ridge regression estimator. The script discusses how machine learning (ML) minimizes the sum of squared differences between the predicted and actual values, which is the essence of linear regression.

💡Ridge Regression

Ridge regression is a form of regression analysis that includes a regularization term to prevent overfitting. It is an extension of ordinary least squares linear regression. The script explains that it adds a penalty term to the loss function, which is the squared norm of the weight vector, scaled by a parameter lambda. This penalty term discourages large values of the coefficients, leading to a more generalized model.

💡Estimator

In statistics and machine learning, an estimator is a rule for estimating an unknown parameter based on observed data. The script introduces two estimators: the ML estimator and the ridge estimator (W hat R). The term 'estimator' is used to describe the method of finding the best-fitting model parameters, such as W in linear regression.

💡Regularization

Regularization is a technique used to prevent overfitting by adding a penalty term to the loss function. The script mentions that the term added to the loss function in ridge regression is a regularizer, which is designed to encourage simpler models that generalize better to unseen data.

💡Hyperparameter

A hyperparameter is a parameter whose value is set prior to the start of the learning process. The script discusses lambda as a hyperparameter in ridge regression, which controls the strength of the regularization. The choice of lambda is typically determined through cross-validation.

💡Prior

In Bayesian statistics, a prior is a probability distribution that expresses the degree of belief in a set of values for an unknown parameter before accounting for new evidence. The script relates the concept of a prior to the preference for certain types of weight vectors in ridge regression, with the Gaussian prior implying a preference for smaller values of W.

💡Norm

In mathematics, the norm is a function that assigns a strictly non-negative length to vectors, and it generalizes the notion of Euclidean distance from a point to the origin. The script uses 'norm W squared' to refer to the penalty term in the ridge regression loss function, which is the squared magnitude of the weight vector.

💡Redundancy

Redundancy in the context of machine learning refers to the presence of correlated or duplicate features that do not provide additional information for the prediction. The script discusses the problem of redundancy in feature sets and how ridge regression can help by penalizing large weights, thus reducing the influence of redundant features.

💡Feature

In machine learning, a feature is an individual measurable property or characteristic used as input for a model. The script uses the terms 'height' and 'weight' as examples of features in a hypothetical regression problem, and it discusses how ridge regression can handle the presence of redundant features.

💡Loss Function

A loss function is a function that maps an event or an outcome to a real number, intuitively representing some 'cost' associated with that event. The script mentions the original loss term in linear regression as the sum of squared differences between the predicted and actual values, and how it is modified in ridge regression by adding a regularization term.

💡Bayesian Modeling

Bayesian modeling is a probabilistic approach to statistical modeling that incorporates prior knowledge or beliefs about parameters into the analysis. The script briefly touches on the idea of putting a prior on the weights (W) in ridge regression, which is a Bayesian viewpoint that influences the selection of the model's parameters.

Highlights

Linear regression can be summarized as minimizing the sum of squared differences between predicted and actual values.

Introduction of a new estimator called ridge regression, which adds a regularization term to the loss function.

The regularization term is a scaled version of the norm of the weights squared, controlled by a hyperparameter lambda.

Ridge regression aims to balance minimizing the loss and avoiding redundancy in the model by penalizing large weight values.

The Bayesian perspective of ridge regression involves putting a prior on the weights, favoring smaller values.

The prior is a Gaussian distribution with zero mean and variance related to the lambda hyperparameter.

The idea behind pushing weights towards zero is to reduce redundancy in features, especially when dealing with a large number of features.

Ridge regression tries to find a balance between a small loss and a small weight norm, avoiding overfitting.

A small lambda value means less penalty for larger weights, while a large lambda value enforces sparser solutions.

The lambda hyperparameter needs to be cross-validated to find the best trade-off between bias and variance.

Ridge regression can be understood geometrically, which may lead to better algorithms or formulations.

The concept of redundancy in features is illustrated with an example involving height, weight, and their linear combinations.

Ridge regression prefers solutions with smaller weight components, even if multiple solutions have the same loss.

The regularization in ridge regression is analogous to pulling the solution towards zero in the weight space.

The strength of the belief in feature redundancy is inversely related to the variance of the prior and directly to lambda.

Ridge regression is a form of regularization that can help in selecting relevant features and reducing overfitting.

The transcript suggests that there may be alternative approaches to regularization beyond ridge regression.

Understanding ridge regression in a geometric context could provide insights into different formulations of the linear regression problem.

Transcripts

 To just see everything together  

from a high level view. So, linear regression  

can be so far whatever we have seen can be  summarized as follows. Linear regression is  

W hat M L is just the arg min over W sum over i  equals 1 to n W transpose x i minus y i squared.  

Now, what we taught is a new estimator, which is  a map estimator, which is usually called W hat  

R and this R stands for ridge, which has roots a  classical statistics. We would not get into that,  

the etymology of this, but it is just arg min  over W for our purpose sum over i equals 1 to  

n W transpose x i minus y i squared, but then  what are we saying we are saying that well,  

you should also add this term second term here,  which is some scaled version of norm W squared.  

So, this is plus some lambda norm W  squared. So, this problem is called as the  

ridge regression problem, this is called as the  ridge regression where you add this extra term  

to your original error or loss. So, this was our  original loss term. And now, you add this extra  

term, which has this Bayesian viewpoint also  or you can think of it as adding this term to  

minimize your mean squared error. So, this term  is what is usually called as a regularizer.  

So, so, basically what we are saying  is that, in addition to the loss,  

you want to add some quantity as a penalty  for somehow preferring some type of W’s.  

So, when I say we put a prior on W and then do a  Bayesian modeling, it means that we are preferring  

some types of W’s. And here the prior if you  remember, what is the prior, the prior was a  

Gaussian with zero mean where zero is a vector  indeed, I mentioned and some variance. So, now,  

that variance term converts to this lambda.  So, this lambda is what is called as a hyper  

parameter, which we had to cross validate,  but then it corresponds to the variance term.  

But what does the prior itself tell us? So, if we say the prior is zero mean, sorry,  

if you say the prior is zero mean, it means that  we are somehow thinking of, we somehow want W’s,  

which are whose length is as small as possible. I  mean, in fact, we want W’s to be all zeros, which  

is a weird thing. So, why would we want that?  Because our prior says that our Gaussian has mean  

zero. So, which means that the maximum probability  is the mode of the distribution is at zero.  

Now, if you think about this, one way to  think about this is that the second term here,  

which comes from the prior the lambda times  norm W squared, is kind of trying to say,  

pull our W the answer towards zero, which  means that it is trying to make the values  

the components of W as small as possible. So,  it is because you are penalizing the length,  

the norm W squared is what is getting penalized,  which is just the length of W itself. So, you want  

it to be as small as possible. You are trying to  pull it towards zero, whereas the first guy is  

trying to make the loss as small as possible. Now, if it so happens, that there are multiple  

W’s which have the same loss, then the W that you  would prefer is the one that has the least length,  

which means that the one that has  many small components. Of course,  

you will not always be able to make W as small as  possible. Of course, you cannot push it to zero,  

zero w is anyway useless. But what is the  idea behind pushing W to zero? The idea behind  

pushing W to zero is that if you have a lot of  features? So, you have, let us say 10,000 or even  

a million features that you are dealing with. Now a lot of these features could be potentially  

redundant. So, maybe you have height, maybe  you have weight, maybe you have two times  

height plus three times weight by somehow you  have a sensor which captures that let us say,  

I am giving a simple example. But then when you  have a lot of features, a lot of redundancies  

could happen. Now, which means that if a linear  combination of these features is what is going to  

represent our label y. So if you have redundant  features, then there might be multiple linear  

combinations which might explain our y label. So, now what we are saying is that pick that  

linear combination, which means pick those set  of weights that have as small length as possible,  

which means that the redundant features, you  are trying to make sure that you do not pick  

multiple redundant features. For example, if  you had height, weight, here is an example.  

So, let us say you have  height, and you have weight.  

And you have 2 times height, plus 3 times  weight, these are your three features.  

And your label, let us say is a noisy version of 3  times height plus 3 times weight. Let us say this  

is your label, label is a noisy version of this.  Now, there are multiple ways you can get this.  

So, one way is to say that, well, I have some  quantity, which is maybe I will say 3 times height  

plus 4 times weight, it is let me use the same  color. 3 times height plus 4 times weight let us  

say, that is our label. Now one way to explain this is to  

say that I put a 1 here, I put a 1 here, and I put  a 1 here. So, this is my weight. For this height,  

this is my weight for this weight, meaning  weightage for the feature weight. And 1 is  

the weightage for the third feature, so this is  f 1, this is f 2, this is f 3. I can have 1, 1,  

1 as my weightage. And then if I add these  three things up, I get 3 times height plus  

4 times weight, which perfectly explains this. But I could also have gotten this by saying that  

this is just 0. And then there is some constant  here and some constant here such that c 1  

times weight, plus c 2 times 2 times height  plus 3 times weight also explains my live.  

So, now here is where I am kind of trying  to avoid this redundancy. Another way to  

do this would be just 2, 3, 4 and 0 here. So, somehow, you might be better off in some  

sense, by unnecessarily avoiding  picking the redundant features.  

So, you want to push as many feature values to  be 0 as possible, assuming that most of them  

are redundant. So, that is the prior assumption  somehow translates to, but then whatever necessary  

features that you should retain such that the loss  can be minimized, you will try to retain that.  

So, that is what this prior is somehow  kind of trying to do that. Now, what is  

this lambda doing the lambda remember,  is something like one by gamma squared,  

where gamma squared was the variance. So, if you  have a very, very strong belief that most of the  

features are redundant, so which means that  there are several W’s close to zero values,  

then we would have very small variance. So, which  means gamma squared would be very, very small, the  

variance is small, which means that most mass is  concentrated around zero according to our prior.  

If gamma squared is very small, 1 by gamma squared  is going to be large, which means lambda will be  

really large, which means that this minimization  is going to pay a large penalty for increasing the  

W. So if you want to choose a W which are larger  weight, larger length, which is norm W squared is  

large, then it means that the amount of penalty  that you are suffering for choosing a larger  

length W is lambda, which is going to be something  like 1 by gamma squared, which will be large,  

if the variance is small, which means that if you  really think that all the weights are redundant,  

then choosing a weight, which is a lot  of large values is less preferred.  

So, your W will not be set something  like that, even if you minimize that.  

So, you can think of these two terms as  balancing how much loss you want versus how  

much redundancy you want to avoid. So, that is  what ridge regression is somehow trying to do.  

Now, one might ask, well, why cannot I try  to directly get as many W values to be zeros  

as possible? Is ridge regression the only way  to do this? Are there other ways to approach  

this regularization? Or, in other words,  how do we understand ridge regression in a  

better way? If I understand it in a better  way, more geometrically, will that lead to  

better algorithms or slightly different types of  formulations of the linear regression problem?  

The answer to all these questions is yes. And  this will lead us to something else, which is  

a different modified version of linear regression.  For that we need to understand ridge regression in  

a slightly different geometry context, which  is what we will start doing next. Thank you.