Gradient Boosting

Gradient Boosting is a powerful machine learning algorithm used for both regression and classification problems. It's an ensemble technique, which means it combines the predictions of several models to improve accuracy and robustness over a single model.

The primary idea behind Gradient Boosting is to add new models to the ensemble sequentially. At each iteration, a new weak, base-learner model is trained with respect to the error of the whole ensemble learnt so far.

The general Gradient Boosting algorithm can be described as follows:

Given a differentiable loss function L(y, F(x)), the objective of Gradient Boosting is to find a model F(x) that minimizes the expected value of this loss function.

We start with an initial model F0(x), which can be as simple as a constant function, for example

F0(x) = argmin(c)E[L(y, c)].

The models F1(x), F2(x), ..., FM(x) are then added sequentially, where each new model is built to predict the residual errors of the current ensemble. The idea is that by doing this iteratively, the overall model can effectively "learn" from the mistakes of the previous models and improve the prediction.

More specifically, at each iteration m, we fit a base learner h(x; a) to the negative gradient of the loss function L(y, F(x)) evaluated at the current model F(x). The parameters of the base learner a are determined by the following least squares problem:

a = argmin(a) Σ L(yi, Fm-1(xi) - h(xi; a))

Here, h(x; a) is a weak learner parameterized by a, and the sum is over the training examples (xi, yi).

Finally, the model is updated by adding the fitted base learner scaled by a learning rate parameter ν (also known as shrinkage parameter) to the current model:

Fm(x) = Fm-1(x) + ν h(x; a)

The learning rate parameter controls the influence of each new base learner, and it's typically set to a small constant (e.g., 0.1) to ensure that the model learns slowly and avoids overfitting.

This process continues for M iterations, where M is a hyperparameter that needs to be chosen by the user. The final model is then given by:

F(x) = F0(x) + ν h(x; a1) + ν h(x; a2) + ... + ν h(x; aM)

It's important to note that the base learners are typically shallow decision trees, but they can also be of other types, such as linear models.

Gradient Boosting requires careful tuning of the learning rate and the number of iterations, as it can otherwise easily overfit the training data. It's also computationally intensive compared to simpler models, such as linear or logistic regression. However, when properly tuned, Gradient Boosting often yields excellent prediction accuracy.