Week 3 Lecture 12 Shrinkage Methods

Summary

The transcript discusses shrinkage methods in regression analysis, focusing on L2 (Ridge) and L1 (Lasso) norm constraints. It explains Lasso's tendency to reduce coefficients to zero, creating a sparse model, contrasting with Ridge's approach to minimize squared error without driving coefficients to zero. The speaker also touches on the concept of derived input directions, which involves creating new feature sets through orthogonalization and dimension reduction for improved regression fits.

Takeaways

• 📉 Shrinkage methods are used to impose constraints on the coefficients (β) to prevent overfitting, such as L2 norm for Ridge Regression and L1 norm for Lasso.
• 🔍 Lasso Regression, which uses L1 norm, is more likely to reduce some coefficients to zero, creating a sparse model, compared to Ridge Regression which tends to shrink coefficients but not eliminate them.
• 🔑 The choice between L1 and L2 norm can depend on prior knowledge about the importance of variables; constraints should be informed to avoid devaluing important variables.
• 📚 Lasso Regression is also known as sparse regression due to its tendency to produce models with many zero coefficients, similar to sparse matrices with few non-zero entries.
• 📐 The geometric interpretation of Lasso constraints suggests that it is more likely to produce solutions at the 'corners' of the feasible region, leading to sparser solutions.
• 🔢 The script discusses the trade-off between minimizing squared error and imposing norm constraints, where Lasso might prefer reducing a large coefficient slightly over reducing a small one to zero.
• 🧩 The transcript mentions the possibility of higher-order norm penalties, such as L4, expanding the range of shrinkage methods available for model regularization.
• 🌐 It introduces a third class of methods for variance reduction that involves derived input directions, moving away from the original set of basis vectors provided.
• 🔄 The process of derived input directions often includes orthogonalization to simplify the regression by allowing univariate regression on each dimension separately.
• 🔍 Dimensionality reduction is part of the derived input directions approach, aiming to find a new set of features that can approximate the original fit with fewer parameters.
• 🛠 The script implies that software packages like R or WEKA can be used to implement Lasso and Ridge Regression, abstracting away much of the computational complexity.

Q & A

What is the primary difference between L2 norm and L1 norm constraints in the context of regression models?

The L2 norm constraint, also known as ridge regression, penalizes the sum of the squares of the coefficients, encouraging smaller values but not necessarily zero. The L1 norm constraint, or lasso regression, penalizes the sum of the absolute values of the coefficients, which can lead to some coefficients being exactly zero, thus performing variable selection.

Why might one choose to use lasso regression over ridge regression?

Lasso regression is preferred when there is a need for feature selection, as it tends to drive some coefficients to zero, effectively selecting a simpler model with fewer features. This can be advantageous in situations where interpretability and model simplicity are important.

What is the concept of 'sparsity' in the context of lasso regression?

Sparsity in lasso regression refers to the tendency of the L1 norm penalty to result in a model with many coefficients being zero. This leads to a sparse model, which is easier to interpret and can generalize better in some cases.

How does the lasso regression handle the scenario where two coefficients can be reduced by the same amount but have different initial values?

Lasso regression is more likely to reduce the coefficient that is closer to zero, as this results in a larger reduction in the penalty term, despite both changes contributing equally to the squared error.

What is the geometric intuition behind why lasso regression is more likely to result in some coefficients being zero?

Geometrically, the lasso constraint forms a diamond shape in the coefficient space, and the solution is more likely to hit a corner of this diamond, which corresponds to some coefficients being exactly zero, as opposed to the circular constraint of ridge regression.

What are derived input directions in the context of regression models?

Derived input directions refer to creating a new set of features or basis vectors through orthogonalization and dimension reduction techniques. This approach transforms the original input space to find a more suitable representation for regression analysis.

Why is orthogonalization of dimensions beneficial in regression models?

Orthogonalization of dimensions allows for univariate regression on each dimension separately, simplifying the model fitting process. Since the dimensions do not interfere with each other, the model can be fit more efficiently.

What is the advantage of using a reduced set of dimensions in regression models?

A reduced set of dimensions can capture the essential information of the original data with fewer variables, leading to a simpler and potentially more robust model that is easier to interpret and less prone to overfitting.

How does the transcript differentiate between subset selection and shrinkage methods in the context of variance reduction?

Subset selection involves explicitly choosing a subset of variables to include in the model, while shrinkage methods, like ridge and lasso regression, adjust the coefficients of all variables but do not necessarily exclude any from the model.

What is the trade-off between minimizing the error and driving coefficients to zero in lasso regression?

In lasso regression, there is a trade-off between minimizing the error and the tendency to drive coefficients to zero. While the model will still aim to minimize the error, it will preferentially reduce coefficients to zero if it can do so without significantly increasing the error.

How does the transcript suggest that one might impose custom constraints on the coefficients based on prior knowledge of the variables?

The transcript suggests that if one has prior knowledge about the importance of certain variables, they can impose custom constraints to ensure that the coefficients of less important variables do not exceed a certain fraction of the coefficients of more important ones.