“A Comprehensive Guide to Huber Regression: Balancing Efficiency and Robustness for Reliable Parameter Estimation

Learn how Huber Regression strikes a balance between ordinary least squares (OLS) and absolute deviation (L1) regression, providing reliable parameter estimates while mitigating the impact of outliers. Understand its advantages, limitations, and practical tips for hyperparameter tuning.

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INTUITION

The HuberRegressor is a robust regression algorithm that combines the advantages of both the least squares method (MSE loss) and the absolute deviation method (MAE loss). It is designed to be less sensitive to outliers compared to ordinary least squares (OLS) regression.

The mathematics behind the HuberRegressor involves a loss function that smoothly transitions from the quadratic loss (MSE) to the linear loss (MAE) based on a threshold parameter called epsilon (ε).

The Huber loss function is defined as follows:

For residuals less than or equal to epsilon: L(residual) = 0.5 * residual^2

For residuals greater than epsilon: L(residual) = epsilon * (abs(residual) - 0.5 * epsilon)

The HuberRegressor minimizes the sum of these Huber loss values to estimate the coefficients that best fit the data.

Here’s a step-by-step overview of how the HuberRegressor algorithm works:

1. Initialize the model parameters, including the coefficient estimates and the epsilon value.
2. Given the input features (X) and target variable (y), calculate the predicted target values (y_pred) using the current coefficient estimates.
3. Calculate the residuals by subtracting the predicted values from the actual target values: residuals = y — y_pred.
4. Based on the residuals, compute the Huber loss for each data point using the defined Huber loss function.
5. Determine the subgradients of the loss function with respect to the coefficients. The subgradient accounts for both the slope and direction…