A Primer on Gradient Boosted Decision Trees

Gradient boosted decision trees are an effective off-the-shelf method for generating effective models for classification and regression tasks.

Gradient boosting is a generic technique that can be applied to arbitrary 'underlying' weak learners - typically decision trees are used. The seminal reference is Friedman's 2001 paper that introduced the method.

Consider a typical supervised learning problem - we are given as input labeled feature vectors $(x, y)$, and seek to estimate a function $\hat F(x)$ that approximates the 'true' mapping $F^\star$, with $F^\star$ minimizing the expected loss $L(y, F(x)$ over some candidate functions $\mathcal{F}$ for a loss function $L$.

In gradient boosting, the model assumes an additive expansion \begin{equation} F(x, \beta, \alpha) = \sum_{i=1}^{n} \beta_{i} h(x, \alpha_{i}) \end{equation} where the $h$ are our weak learners. Thus, the predictor from gradient boosting is a linear combination of weak learners, and the procedure does two things:

  • Computes $\beta_m$ - the weight that a given classifier has in context of the ensemble.
  • Weights the training examples to compute the $i$-th weak classifier $h(\cdot, \alpha_m)$.

The Algorithm

For examples, we'll use the decision tree training library I wrote in Go, available on GitHub at https://github.com/ajtulloch/decisiontrees.

The boosting algorithm, in pseudo-code, is quite simple:

  • initialize list of weak learners to a singleton list with simple prior
  • for each round in 1..numRounds:
  • reweight examples $(x, y)$ to $(x, \tilde y)$ by 'upweighting' examples that the existing forest poorly predicts
  • estimate new weak classifier $h_i$ on weighted examples
  • compute weight $\beta_i$ of new weak classifier
  • add the pair $(h_i, \beta_i)$ to the forest
  • return forest

The intuition behind gradient boosting is quite simple - we iteratively build a sequence of predictors, and our final predictor is a weighted average of these predictors. At each step, we focus on adding an incremental classifier that improves the performance of the entire ensemble. The technical description is 'gradient descent in functional space'.

Thus, if we have examples that are not well predicted by the current ensemble, the next stage will work harder to fit these examples.

Example Implementation

For an implementation of the above approach, see the boosting.go file on GitHub. The loop body is a simple function inlined below:

There are some complications (stochastic gradient boosting, influence trimming), although the core algorithm is as described in the pseudocode.

In subsequent posts we'll elaborate on

  • Algorithms for training the individual weak learners
  • Boosting and decision tree hyperparameters
  • Speeding up training and prediction
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