Random Forests for Big Data

Big Data is one of the major challenges of statistical science and has numerous consequences from algorithmic and theoretical viewpoints. Big Data always involve massive data but they also often include online data and data heterogeneity. Recently some statistical methods have been adapted to process Big Data, like linear regression models, clustering methods and bootstrapping schemes. Based on decision trees combined with aggregation and bootstrap ideas, random forests were introduced by Breiman in 2001. They are a powerful nonparametric statistical method allowing to consider in a single and versatile framework regression problems, as well as two-class and multi-class classification problems. Focusing on classification problems, this paper proposes a selective review of available proposals that deal with scaling random forests to Big Data problems. These proposals rely on parallel environments or on online adaptations of random forests. We also describe how related quantities -- such as out-of-bag error and variable importance -- are addressed in these methods. Then, we formulate various remarks for random forests in the Big Data context. Finally, we experiment five variants on two massive datasets (15 and 120 millions of observations), a simulated one as well as real world data. One variant relies on subsampling while three others are related to parallel implementations of random forests and involve either various adaptations of bootstrap to Big Data or to "divide-and-conquer" approaches. The fifth variant relates on online learning of random forests. These numerical experiments lead to highlight the relative performance of the different variants, as well as some of their limitations

Distributed Robust Learning

We propose a framework for distributed robust statistical learning on {\em big contaminated data}. The Distributed Robust Learning (DRL) framework can reduce the computational time of traditional robust learning methods by several orders of magnitude. We analyze the robustness property of DRL, showing that DRL not only preserves the robustness of the base robust learning method, but also tolerates contaminations on a constant fraction of results from computing nodes (node failures). More precisely, even in presence of the most adversarial outlier distribution over computing nodes, DRL still achieves a breakdown point of at least $\lambda^*/2$, where $\lambda^*$ is the break down point of corresponding centralized algorithm. This is in stark contrast with naive division-and-averaging implementation, which may reduce the breakdown point by a factor of $k$ when $k$ computing nodes are used. We then specialize the DRL framework for two concrete cases: distributed robust principal component analysis and distributed robust regression. We demonstrate the efficiency and the robustness advantages of DRL through comprehensive simulations and predicting image tags on a large-scale image set.Comment: 18 pages, 2 figure

Distributed Low-rank Subspace Segmentation

Vision problems ranging from image clustering to motion segmentation to semi-supervised learning can naturally be framed as subspace segmentation problems, in which one aims to recover multiple low-dimensional subspaces from noisy and corrupted input data. Low-Rank Representation (LRR), a convex formulation of the subspace segmentation problem, is provably and empirically accurate on small problems but does not scale to the massive sizes of modern vision datasets. Moreover, past work aimed at scaling up low-rank matrix factorization is not applicable to LRR given its non-decomposable constraints. In this work, we propose a novel divide-and-conquer algorithm for large-scale subspace segmentation that can cope with LRR's non-decomposable constraints and maintains LRR's strong recovery guarantees. This has immediate implications for the scalability of subspace segmentation, which we demonstrate on a benchmark face recognition dataset and in simulations. We then introduce novel applications of LRR-based subspace segmentation to large-scale semi-supervised learning for multimedia event detection, concept detection, and image tagging. In each case, we obtain state-of-the-art results and order-of-magnitude speed ups

Online Updating of Statistical Inference in the Big Data Setting

We present statistical methods for big data arising from online analytical processing, where large amounts of data arrive in streams and require fast analysis without storage/access to the historical data. In particular, we develop iterative estimating algorithms and statistical inferences for linear models and estimating equations that update as new data arrive. These algorithms are computationally efficient, minimally storage-intensive, and allow for possible rank deficiencies in the subset design matrices due to rare-event covariates. Within the linear model setting, the proposed online-updating framework leads to predictive residual tests that can be used to assess the goodness-of-fit of the hypothesized model. We also propose a new online-updating estimator under the estimating equation setting. Theoretical properties of the goodness-of-fit tests and proposed estimators are examined in detail. In simulation studies and real data applications, our estimator compares favorably with competing approaches under the estimating equation setting.Comment: Submitted to Technometric