Least-mean squares (LMS) solvers such as Linear / Ridge / Lasso-Regression, SVD and Elastic-Net not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as decision trees and matrix factorizations. We suggest an algorithm that gets a finite set of $n$ $d$-dimensional real vectors and returns a weighted subset of $d+1$ vectors whose sum is \emph{exactly} the same. The proof in Caratheodory's Theorem (1907) computes such a subset in $O(n^2d^2)$ time and thus not used in practice. Our algorithm computes this subset in $O(nd)$ time, using $O(\log n)$ calls to Caratheodory's construction on small but "smart" subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets. As an example application, we show how it can be used to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to x100. Generalization for streaming and distributed (big) data is trivial. Extensive experimental results and complete open source code are also provided.

In safety-critical applications a probabilistic model is usually required to be calibrated, i.e., to capture the uncertainty of its predictions accurately. In multi-class classification, calibration of the most confident predictions only is often not sufficient. We propose and study calibration measures for multi-class classification that generalize existing measures such as the expected calibration error, the maximum calibration error, and the maximum mean calibration error. We propose and evaluate empirically different consistent and unbiased estimators for a specific class of measures based on matrix-valued kernels. Importantly, these estimators can be interpreted as test statistics associated with well-defined bounds and approximations of the p-value under the null hypothesis that the model is calibrated, significantly improving the interpretability of calibration measures, which otherwise lack any meaningful unit or scale.

Applications such as weather forecasting and personalized medicine demand models that output calibrated probability estimates---those representative of the true likelihood of a prediction. Most models are not calibrated out of the box but are recalibrated by post-processing model outputs. We find in this work that popular recalibration methods like Platt scaling and temperature scaling are (i) less calibrated than reported, and (ii) current techniques cannot estimate how miscalibrated they are. An alternative method, histogram binning, has measurable calibration error but is sample inefficient---it requires $O(B/\epsilon^2)$ samples, compared to $O(1/\epsilon^2)$ for scaling methods, where $B$ is the number of distinct probabilities the model can output. To get the best of both worlds, we introduce the scaling-binning calibrator, which first fits a parametric function that acts like a baseline for variance reduction and then bins the function values to actually ensure calibration. This requires only $O(1/\epsilon^2 + B)$ samples. We then show that methods used to estimate calibration error are suboptimal---we prove that an alternative estimator introduced in the meteorological community requires fewer samples ($O(\sqrt{B})$ instead of $O(B)$). We validate our approach with multiclass calibration experiments on CIFAR-10 and ImageNet, where we obtain a 35\% lower calibration error than histogram binning and, unlike scaling methods, guarantees on true calibration.

Estimating graphical model structure from high-dimensional and undersampled data is a fundamental problem in many scientific fields. Existing approaches, such as GLASSO, latent variable GLASSO, and latent tree models, suffer from high computational complexity and may impose unrealistic sparsity priors in some cases. We introduce a novel method that leverages a newly discovered connection between information-theoretic measures and structured latent factor models to derive an optimization objective which encourages modular structures where each observed variable has a single latent parent. The proposed method has linear stepwise computational complexity w.r.t. the number of observed variables. Our experiments on synthetic data demonstrate that our approach is the only method that recovers modular structure better as the dimensionality increases. We also use our approach for estimating covariance structure for a number of real-world datasets and show that it consistently outperforms state-of-the-art estimators at a fraction of the computational cost. Finally, we apply the proposed method to high-resolution fMRI data (with more than 10^5 voxels) and show that it is capable of extracting meaningful patterns.

Given a n-by-d data matrix A, principal component projection (PCP) and principal component regression (PCR), i.e. projection and regression restricted to the top-eigenspace of A, are fundamental problems in machine learning, optimization, and numerical analysis. In this paper we provide the first algorithms that solve these problems in nearly linear time for fixed eigenvalue distribution and large n. This improves upon previous methods which had superlinear running times when either the number of top eigenvalues or gap between the eigenspaces were large.

We achieve our results by applying rational polynomial approximations to reduce the problem to solving asymmetric linear systems which we solve by a variant of SVRG. We corroborate these findings with preliminary empirical experiments.

We consider the problem of detecting anomalies in a large dataset. We propose a framework called Partial Identification which captures the intuition that anomalies are easy to distinguish from the overwhelming majority of points by relatively few attribute values. Formalizing this intuition, we propose a geometric anomaly measure for a point that we call PIDScore, which measures the minimum density of data points over all subcubes containing the point. We present PIDForest: a random forest based algorithm that finds anomalies based on this definition. We show that it performs favorably in comparison to several popular anomaly detection methods, across a broad range of benchmarks. PIDForest also provides a succinct explanation for why a point is labelled anomalous, by providing a set of features and ranges for them which are relatively uncommon in the dataset.