# Online passive-aggressive algorithms

One of my favourite papers is the paper on online passive-aggressive algorithms [1] by Koby Crammer et al. In this paper, a set of algorithms is presented, that can be used to learn classifiers and regressors with a simple, closed form update. The underlying intuition is that when a prediction is good enough, the model is not updated (i.e. it is passive). When the prediction is not good enough, the model is aggressively updated not to make that mistake again. This is done after every data point. Extensions are presented for multi-class predictions, one-class prediction and structured prediction.

What I like so much about the paper is that is is very clear about the objective of the algorithm, and that the derivation of the update is straight forward to follow. It results in a set of algorithms (with regret bounds!) that is very easy to implement, even in low-level programming languages. A side effect of the online update is no additional state needs to be tracked nor stored.

I have used passive-aggressive regression (PA-I), for example for an online learning regressor on an embedded system. The fact that the online update is numerically stable, and never diverges makes it a very robust tool. Contrast that with gradient descent, where we have to be careful with learning rates.

One can implicitly do a non-linear feature mappings in the passive aggressive algorithms with the kernel-trick. This transforms the passive aggressive models into non-linear model, not unlike a support vector machine. Wang and Vucetic show in [1] how to limit the number of support vectors to a small budget, so that it still can be used in an online fashion. This extension is also easy to implement in low-level languages, such as C.

1. Crammer, Koby, et al. “Online passive-aggressive algorithms.” Journal of Machine Learning Research 7.Mar (2006): 551-585.
2. Wang, Zhuang and Slobodan Vucetic. “Online Passive-Aggressive Algorithms on a Budget.” AISTATS (2010).