As predictive models become more expressive, they also become more vulnerable to a common problem: overfitting. A model that fits training data too closely may capture noise rather than underlying patterns, leading to poor performance on unseen data. In linear models, this issue often arises when the number of features is large or when features are highly correlated. Regularisation techniques provide a mathematically grounded way to control model complexity, striking a balance between accuracy and generalisation. Among these techniques, L1 (Lasso) and L2 (Ridge) regularisation are foundational tools that every data practitioner should understand.
Why Model Complexity Needs Control
Linear models are deceptively simple. Although their structure is straightforward, their behaviour can become unstable as coefficients increase. Large weights allow the model to fit subtle fluctuations in the training data, but these fluctuations rarely persist in real-world scenarios.
Regularisation addresses this by incorporating a penalty on model complexity directly into the model’s objective function. Instead of minimising prediction error alone, the model is encouraged to find solutions that are both accurate and restrained. This controlled learning process leads to more stable coefficients, improved generalisation, and greater robustness to noisy data. These concepts are often explored early in a data science course in mumbai, where theoretical understanding is linked closely with practical modelling challenges.
L2 Regularisation (Ridge): Penalising Large Coefficients
Ridge regression, or L2 regularisation, modifies the standard least squares objective by adding a penalty proportional to the sum of the squared coefficients. Mathematically, the objective function includes an additional term that discourages large weights without forcing them to zero.
This squared penalty has a smoothing effect. Coefficients are shrunk towards zero, but rarely eliminated entirely. As a result, Ridge regression is particularly effective when dealing with multicollinearity, where features are correlated with each other. Instead of arbitrarily assigning large positive or negative weights, Ridge distributes influence more evenly across related features.
From an optimisation perspective, L2 regularisation keeps the solution space well-behaved. The objective function remains differentiable, facilitating solution via gradient-based methods. In practice, Ridge regression often improves predictive performance when all features contribute some signal, even if that signal is weak.
L1 Regularisation (Lasso): Enforcing Sparsity
Lasso regression, or L1 regularisation, introduces a penalty based on the absolute values of the coefficients. This seemingly small change has a significant impact on model behaviour. The L1 penalty encourages sparsity, meaning that some coefficients are driven exactly to zero.
This property makes Lasso a powerful tool for feature selection. By eliminating irrelevant or redundant features, it produces simpler, more interpretable models. In high-dimensional datasets, where the number of features may exceed the number of observations, Lasso helps reduce complexity while retaining predictive power.
However, this sparsity comes with trade-offs. When features are highly correlated, Lasso may select one arbitrarily and discard others, which can lead to instability. Understanding when to apply L1 regularisation, and how to tune its strength, is a key skill developed through hands-on learning and experimentation.
Choosing Between L1 and L2 Regularization
The choice between Lasso and Ridge depends on the nature of the problem and the goals of the analysis. If interpretability and feature selection are priorities, L1 regularisation is often preferred. If the goal is to stabilise predictions while retaining all features, L2 regularisation is usually more suitable.
In practice, many practitioners use a combination of both through elastic net regularisation. This approach combines L1 and L2 penalties, balancing sparsity and stability. It is especially useful in datasets with many correlated features, where neither Lasso nor Ridge alone performs optimally.
Learning how to make these choices requires both theoretical understanding and practical intuition. Programmes such as a data science course in mumbai often emphasise this decision-making process by exposing learners to real datasets and evaluation scenarios.
The Role of Regularisation Strength
Both L1 and L2 regularisation introduce a hyperparameter that controls the strength of the penalty. A small value results in behaviour close to ordinary least squares, while a considerable value enforces stronger constraints on the coefficients.
Selecting the appropriate regularisation strength is critical. Too little regularisation fails to prevent overfitting. Excessive complexity leads to underfitting, in which the model becomes overly simplistic. Techniques such as cross-validation are commonly used to find a balance that minimises validation error while maintaining generalisation.
This tuning process highlights an essential principle in machine learning. Regularisation is not about suppressing complexity blindly, but about guiding the model towards solutions that generalise well beyond the training data.
Conclusion
Regularisation techniques play a central role in controlling model complexity and preventing overfitting in linear models. L2 regularisation stabilises solutions by shrinking coefficients, while L1 regularisation promotes sparsity and interpretability. Understanding their mathematical foundations and practical implications allows practitioners to build models that are both accurate and robust. By applying these techniques thoughtfully and tuning them carefully, data scientists can ensure that their models capture meaningful patterns rather than noise, leading to more reliable predictions in real-world applications.