Nathan Srebro

Pegasos: Primal Estimated sub-GrAdient SOlver for SVM

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 ε is Õ(1/ε). In contrast, previous analyses of stochastic gradient descent methods require Ω (1/ε2) iterations. As in previously devised SVM solvers, the number of iterations also scales linearly with 1/λ, where λ is the regularization parameter of SVM. For a linear kernel, the total run-time of our method is Õ (d/(λε)), where d is a bound on the number of non-zero features in each example. Since the run-time does 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.

Fast maximum margin matrix factorization for collaborative prediction

Maximum Margin Matrix Factorization (MMMF) was recently suggested (Srebro et al., 2005) as a convex, infinite dimensional alternative to low-rank approximations and standard factor models. MMMF can be formulated as a semi-definite programming (SDP) and learned using standard SDP solvers. However, current SDP solvers can only handle MMMF problems on matrices of dimensionality up to a few hundred. Here, we investigate a direct gradient-based optimization method for MMMF and demonstrate it on large collaborative prediction problems. We compare against results obtained by Marlin (2004) and find that MMMF substantially outperforms all nine methods he tested.

Pegasos: primal estimated sub-gradient solver for SVM

We describe and analyze a simple and effective stochastic sub-gradient descent algorithm for solving the optimization problem cast by Support Vector Machines (SVM). We prove that the number of iterations required to obtain a solution of accuracy

SVM optimization: inverse dependence on training set size

{\epsilon}

Stochastic Gradient Descent, Weighted Sampling, and the Randomized Kaczmarz algorithm

is

Maximum likelihood bounded tree-width Markov networks

{\tilde{O}(1 / \epsilon)}