Pegasos: Primal Estimated sub-GrAdient SOlver for SVM
Shai Shalev-Shwartz, Yoram Singer and Nathan Srebro
In: ICML 2007(2007).

## Abstract

We describe and analyze a simple and effective iterative algorithm for solving the optimization problem cast by Support Vector Machines (SVM). Our method alternates between stochastic gradient descent steps and projection steps. We prove that the number of iterations required to obtain a solution of accuracy $\epsilon$ is $\tilde{O}(1 / \epsilon)$. In contrast, previous analyses of stochastic gradient descent methods require $\Omega(1 / \epsilon^2)$ iterations. As in previously devised SVM solvers, the number of iterations also scales linearly with $1/\lambda$, where $\lambda$ is the regularization parameter of SVM. For a linear kernel, the total run-time of our method is $\tilde{O}(d/(\lambda \epsilon))$, where $d$ is a bound on the number of non-zero features in each example. Since the run-time does {\em not} depend directly on the size of the training set, the resulting algorithm is especially suited for learning from large datasets. Our approach can seamlessly be adapted to employ non-linear kernels while working solely on the primal objective function. We demonstrate the efficiency and applicability of our approach by conducting experiments on large text classification problems, comparing our solver to existing state-of-the-art SVM solvers. For example, it takes less than $5$ seconds for our solver to converge when solving a text classification problem from Reuters Corpus Volume 1 (RCV1) with $800,000$ training examples.