Compare L1 and L2 regularization and their impact on model coefficients.

Compare L1 and L2 Regularization and Their Impact on Model Coefficients

In Data Science, one of the common challenges students face is overfitting — when a model learns noise in the training data instead of generalizable patterns. Regularization is a powerful tool to control this. Two popular types are L1 regularization (Lasso) and L2 regularization (Ridge). In this post, we explore what they are, how they differ, how they affect model coefficients (with statistics), and why in our Data Science Course, we teach them carefully under the principle of Quality Thought — i.e. thinking deeply about trade-offs and interpretability.

What are L1 and L2 Regularization?

L1 (Lasso) adds a penalty equal to the sum of absolute values of the coefficients to the loss function. It tends to push some coefficients exactly to zero, effectively performing feature selection.
L2 (Ridge) adds a penalty equal to the sum of squares of coefficients. It shrinks all coefficients toward zero but usually none becomes exactly zero.

Both have a hyperparameter (often called λ or α) controlling penalty strength. As λ increases, regularization becomes stronger. Too small → overfitting; too large → underfitting.

Impact on Model Coefficients — With Stats

To illustrate statistical behaviour:

From simulations (explained.ai), when sampling random quadratic loss functions, ≈ 66% of trials with L1 regularization gave at least one coefficient exactly zero, compared to only ~3% under L2 for symmetric loss functions. For more general asymmetric or rotated loss functions, L1 gave zero coefficients in ~72% of the trials vs ~5% for L2.
In real‐world logistic regression experiments (e.g. in “An Experiment on Feature Selection using Logistic Regression”, 2024), L1 regularization allowed ranking features by nonzero coefficients and selecting smaller feature sets without significant loss of accuracy compared to L2.

So statistically, L1 is much more likely to produce sparse coefficient vectors (many zeros), which makes the model more interpretable and simpler. L2 tends instead to produce dense vectors but with smaller magnitude coefficients.

Trade-offs and When to Use Which

Sparsity & Interpretability: If you want a simpler model, i.e. fewer features, L1 is better.
Multicollinearity: If features are highly correlated, L2 tends to distribute weights among them, giving smaller but nonzero coefficients; L1 may arbitrarily pick one and zero others.
Stability vs Feature Selection: L2 tends to give more stable solutions; small perturbations in data or λ don’t usually make coefficients zero. L1 is more sensitive.
Computational Considerations: Because L1 involves absolute values and is not differentiable at 0, its optimization often requires specialized methods like coordinate descent or sub-gradient methods. L2 is smoother, optimization is more straightforward.

Quality Thought: How to Help You Learn This Right

In our Data Science Course, we emphasize Quality Thought by:

Not just teaching “how”, but why L1 and L2 behave differently in terms of geometry (constraint shapes), bias-variance trade-off, coefficient paths as λ changes.
Showing empirical results: plot coefficient paths for Lasso/Ridge with increasing λ, so students see which features drop out (L1) vs. which shrink (L2).
Guiding on hyperparameter tuning (cross-validation) so students don’t pick λ by guesswork.
Helping with feature scaling (standardization), since without scaling, regularization penalizes differently for features with different units / magnitudes.

Example in Practice

Suppose you have a dataset with 100 features, many of which are noise or irrelevant. Using L1 regularization with a suitably chosen λ may reduce the model to 5-10 nonzero coefficients, making the model easier to interpret and faster to compute. Using L2 instead will keep all 100 but shrink many towards zero. Depending on your goal (interpretability vs predictive accuracy vs stability), you choose accordingly.

A published simulation found that for symmetric loss functions, about 66% of L1 regularized models had zero coefficients, whereas only 3% under L2 did. This shows a statistical likelihood of sparsity under L1.

How Our Courses Help Educational Students

We ensure students not only understand the theoretical aspects of regularization, but also get hands-on practice. In our courses:

Labs where you apply L1 and L2 regularization on real datasets, see coefficient effects.
Assignments where you compare Lasso vs Ridge vs Elastic Net.
Guidance on reading research papers (e.g., “Experiment on Feature Selection using Logistic Regression” 2024) to see how leading statisticians design experiments and interpret results.
Emphasis on Quality Thought, so students learn to ask: Which regularizer is appropriate? What is the cost of interpretability vs predictive performance? How to avoid underfitting/overfitting?

Conclusion

L1 and L2 regularization are foundational tools in a data scientist’s toolkit. L1 tends to yield sparse, interpretable models by driving many coefficients exactly to zero; L2 tends to shrink all coefficients, keeping them nonzero but more modest in magnitude. Choosing between them involves trade-offs: interpretability vs stability, feature selection vs preserving correlated predictors, simplicity vs predictive power. With strong understanding and practice, students can decide which regularization suits a given problem. With Quality Thought ingrained, our courses aim to equip Educational Students in the Data Science Course with both theory and practical experience—so that when faced with a dataset, a model, and the decision of which regularization to apply, they can choose wisely rather than by accident?

Explain how Support Vector Machines handle non-linearly separable data.

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