## Understanding Regularization in Machine Learning

Mitigating overfitting is a crucial step in fine-tuning your Machine Learning model. Overfitting, detrimental to accuracy, happens when a model overly focuses on incidental data in the training set rather than the underlying pattern. This incidental data, known as noise, represents anomalies in your dataset rather than key attributes.

## Role of Regularization in Regression

In machine learning, this pitfall is curtailed through regularization in regression, where estimates of coefficients are restrained or 'shrunken' towards zero. The underlying goal is to prevent the model from becoming extremely complex and overfitting the data, keeping it simple and controlled.

The relation with regularized regression is denoted as Y for learned estimations, and (X) for coefficient estimations for varying variables or predictors. The Residual Sum of Squares (RSS) serves as the loss function in the fitting process, aiming to minimize this function for the best coefficients. When the training data includes incidental data or noise, the calculated coefficients are unlikely to adapt well to subsequent data. Here, regularization comes into play by reducing or regularizing these learned estimations towards zero.

## Lasso and Ridge Regression: A Closer Look

### Ridge Regression

In Lasso regression, high coefficients are penalized by leveraging the modulus instead of squares of β, this penalty method being referred to as the L1 norm. Whereas Ridge regression restrains the sum of squares of coefficients within a specified constraint, s.

These constraint functions are used to establish parameters. For instance, Ridge regression can be written as �12+�22≤�*β*12+*β*22≤*s*, where all points within the radius defined by �12+�22≤�*β*12+*β*22≤*s* have the least loss function.

By adjusting the tuning value, the impact of Ridge regression can be controlled. When it is set to zero, the Ridge regression estimates are identical to those of the least squares. However, as the tuning parameter nears zero, the shrinkage penalty becomes more consequential and the Ridge regression coefficient estimates plummet towards zero.

### Lasso Regression

Likewise, the Lasso equation would manifest as ∣�1∣+∣�2∣≤�∣*β*1∣+∣*β*2∣≤*s*, implying that positions within a diamond-shaped boundary set by ∣�1∣+∣�2∣≤�∣*β*1∣+∣*β*2∣≤*s* have the minimum loss function.

However, the catch with Ridge regression is that it lacks scale equivariance and also impacts the model interpretability. The least significant predictors' coefficients are diminished close to zero, but never to the absolute zero. This means all predictors remain in the final model.

In contrast, Lasso's L1 penalty drives multiple coefficient estimates to absolute zero when the tuning parameter is large, leading to variable selection. The lasso technique produces sparse models and makes variable selection achievable.

## The Balance between Bias and Variance

Regularization in machine learning ensures a balance between bias and variance, backed by the tuning parameter. As the tuning parameter increases, the coefficients decline, leading to lower variance. This diminishes overfitting and the loss of critical properties within the data. Nevertheless, if the tuning parameter is excessively high, the model suffers from bias and underfitting.