UPDATE: I wrote this post a long time ago and it hasn't been very informative. I have learned more about boosting and an updated concise description of the relationship between the two can be found here.

I’d like to share my current understanding of this question after some reading:

1. Friedman, Jerome; Hastie, Trevor; Tibshirani, Robert. Additive logistic regression: a statistical view of boosting (With discussion and a rejoinder by the authors). Ann. Statist. 28 (2000), no. 2, 337–407. doi:10.1214/aos/1016218223. http://projecteuclid.org/euclid.aos/1016218223.
2. Friedman, Jerome H. Greedy function approximation: A gradient boosting machine. Ann. Statist. 29 (2001), no. 5, 1189–1232. doi:10.1214/aos/1013203451. http://projecteuclid.org/euclid.aos/1013203451.
3. Chapter 10. Boosting and Additive Trees. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Second Edition, February 2009, Trevor Hastie, Robert Tibshirani, and Jerome Friedman

They are all from the same authors. The first two are very technically detailed, while the book chapter provides a relatively more high-level view.

TL;DR:

1. The two are concepts not at the same level, hence their comparison are not so meaningful.
2. Two more higher level concepts are needed to explain the relationship between them:
1. Forward stagewise additive modeling (FSAM)

Basically,

1. FSAM is a specific form of additive model
2. Adaboost is a specific form of FSAM
3. Gradient boosting, on the other hand, is a numerical process for optimizing FSAM using gradient descent by treating functions as numerical parameters.
4. Adaboost, given it is a two-class classification problem with scaled classification trees (with $-1/1$ encoding for classes), and a specific loss function (exponent loss), the opimization of its coressponding FSAM is relatively easy even without gradient descent.

More:

Let’s start with additive modeling first, which is effectively the ensemble method, i.e. combining how multiple models ($M$) together.

\begin{equation} f(x) = \sum_{m=1}^{M} \beta_{m}b(x;\gamma_m) \label{eq:additiveModel} \end{equation}

where

• $\beta_m$ is the associated coefficent (also named expansion coefficient in the aforementioned book chapter), and
• $b(x;\gamma_m$) is the $m$th basis function (e.g. tree stump) with
• $x$ being the input data, and
• $\gamma_m$ being the model parameters (e.g. split variable and cutoff for a tree stump).

\begin{equation} f_m(x) = f_{m-1}(x) + \beta b(x;\gamma) \label{eq:fsam} \end{equation}

where $f_m(x)$ is the learned model at the $m$th iteration (e.g. boosting iteration), and there could be $M$ iterations.

However, its sepcialty is that there is an element of sequence dependency in it, hence the so-called forward stagewise, i.e. at iteration $m$, $f_m$ depends on $f_{m-1}$. Noteably, the parameters of the previously added basis functions won’t be adjusted in later iterations. The advantage of such a dependency is that it makes the model optimization (learning) process easier. I still have no clue how to optimize the additive model \eqref{eq:additiveModel} altogether at once, considering hundreds of thousands of $b$ (e.g. $M=1000$).

Now, we get to AdaBoost, which is a specific case of FSAM, with an exponential loss function:

\begin{equation} L(y, f(x)) = e^{(-y f(x))} \label{eq:exponentialLoss} \end{equation}

The is not obvious at all without looking at the algorithm of Adaboost. But even after so, it’s not so intuitive. That *Adaboost is a type of FSAM was not discovered till 5 years later it was first invented. After all, it’s proved in the book chapter (I also rederived it on paper) that by plugging in \eqref{eq:exponentialLoss}, adaboost fits into a specific case of \eqref{eq:fsam}.

Before we get to gradient boosting, let’s look at the exponent of the loss function $(y f(x))$ with negative sign removed. This is called the “margin”, and interestingly it has a analogous role to the residual $y - f(x)$ in regression. For formalize it a bit the two expressions are analogous:

• $y \cdot f(x)$ in a classification problem with ($-1/1$ response)
• $y - f(x)$ in a regression problem:

where $y$ is the truth label and $f(x)$ is the model prediction. Concretely, when the $y f(x)$ is $1$ when the classification is correct otherwise $-1$.

Now, let’s get to gradient boosting, which describes a numerical method for optimizing FSAM using gradient descent.

TODO: will update with more details on gradient boosting later.

Thank you for reading along. Please leavage a message and correct me if my understanding is not correct.