We consider learning in the presence of arbitrary noise that can overwhelm the signal in terms of magnitude on a fraction of data points observed (aka outliers). Standard approaches based on minimizing empirical loss can fail miserably and lead to arbitrary bad solutions in this setting. We propose an approach that iterates between finding a solution with minimal empirical loss and re-weighting the data, reinforcing data points where the previous solution works well. We show that our approach can handle arbitrarily large noise, is robust as having a non-trivial breakdown point, and converges linearly under certain conditions. The intuitive idea of our approach is to automatically exclude “difficult” data points from model fitting. More importantly (and perhaps surprisingly), we validate this intuition by establishing guarantees for generalization and iteration complexity that \em essentially ignore the presence of outliers

Abstract We propose a family of non-uniform sampling strategies to provably speed up a class of stochastic optimization algorithms with linear convergence including Stochastic Variance Reduced Gradient (SVRG) and Stochastic Dual Coordinate Ascent (SDCA). For a large family of penalized empirical risk minimization problems, our methods exploit data dependent local smoothness of the loss functions near the optimum, while maintaining convergence guarantees. Our bounds are the first to quantify the advantage gained from local smoothness which are significant for some problems significantly better. Empirically, we provide thorough numerical results to back up our theory. Additionally we present algorithms exploiting local smoothness in more aggressive ways, which perform even better in practice.

We consider the general problem of Multiple Model Learning (MML) from data, from the statistical and algorithmic perspectives; this problem includes clustering, multiple regression and subspace clustering as special cases. A common approach to solving new MML problems is to generalize Lloyd's algorithm for clustering (or Expectation-Maximization for soft clustering). However this approach is unfortunately sensitive to outliers and large noise: a single exceptional point may take over one of the models. We propose a different general formulation that seeks for each model a distribution over data points; the weights are regularized to be sufficiently spread out. This enhances robustness by making assumptions on class balance. We further provide generalization bounds and explain how the new iterations may be computed efficiently. We demonstrate the robustness benefits of our approach with some experimental results and prove for the important case of clustering that our approach has a non-trivial breakdown point, i.e., is guaranteed to be robust to a fixed percentage of adversarial unbounded outliers.