Robust classification via MOM minimization

We present an extension of Vapnik's classical empirical risk minimizer (ERM) where the empirical risk is replaced by a median-of-means (MOM) estimator, the new estimators are called MOM minimizers. While ERM is sensitive to corruption of the dataset for many classical loss functions used in classification, we show that MOM minimizers behave well in theory, in the sense that it achieves Vapnik's (slow) rates of convergence under weak assumptions: data are only required to have a finite second moment and some outliers may also have corrupted the dataset. We propose an algorithm inspired by MOM minimizers. These algorithms can be analyzed using arguments quite similar to those used for Stochastic Block Gradient descent. As a proof of concept, we show how to modify a proof of consistency for a descent algorithm to prove consistency of its MOM version. As MOM algorithms perform a smart subsampling, our procedure can also help to reduce substantially time computations and memory ressources when applied to non linear algorithms. These empirical performances are illustrated on both simulated and real datasets

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view

Petuum: A New Platform for Distributed Machine Learning on Big Data

What is a systematic way to efficiently apply a wide spectrum of advanced ML programs to industrial scale problems, using Big Models (up to 100s of billions of parameters) on Big Data (up to terabytes or petabytes)? Modern parallelization strategies employ fine-grained operations and scheduling beyond the classic bulk-synchronous processing paradigm popularized by MapReduce, or even specialized graph-based execution that relies on graph representations of ML programs. The variety of approaches tends to pull systems and algorithms design in different directions, and it remains difficult to find a universal platform applicable to a wide range of ML programs at scale. We propose a general-purpose framework that systematically addresses data- and model-parallel challenges in large-scale ML, by observing that many ML programs are fundamentally optimization-centric and admit error-tolerant, iterative-convergent algorithmic solutions. This presents unique opportunities for an integrative system design, such as bounded-error network synchronization and dynamic scheduling based on ML program structure. We demonstrate the efficacy of these system designs versus well-known implementations of modern ML algorithms, allowing ML programs to run in much less time and at considerably larger model sizes, even on modestly-sized compute clusters.Comment: 15 pages, 10 figures, final version in KDD 2015 under the same titl