Generalized linear models (GLMs)---such as logistic regression, Poisson regression, and robust regression---provide interpretable models for diverse data types. Probabilistic approaches, particularly Bayesian ones, allow coherent estimates of uncertainty, incorporation of prior information, and sharing of power across experiments via hierarchical models. In practice, however, the approximate Bayesian methods necessary for inference have either failed to scale to large data sets or failed to provide theoretical guarantees on the quality of inference. We propose a new approach based on constructing polynomial approximate sufficient statistics for GLMs (PASS-GLM). We demonstrate that our method admits a simple algorithm as well as trivial streaming and distributed extensions that do not compound error across computations. We provide theoretical guarantees on the quality of point (MAP) estimates, the approximate posterior, and posterior mean and uncertainty estimates. We validate our approach empirically in the case of logistic regression using a quadratic approximation and show competitive performance with stochastic gradient descent, MCMC, and the Laplace approximation in terms of speed and multiple measures of accuracy---including on an advertising data set with 40 million data points and 20,000 covariates.

Estimation in generalized linear models (GLM) is complicated by the presence of constraints. One can handle constraints by maximizing a penalized log-likelihood. Penalties such as the lasso are effective in high dimensions but often lead to severe shrinkage. This paper explores instead penalizing the squared distance to constraint sets. Distance penalties are more flexible than algebraic and regularization penalties, and avoid the drawback of shrinkage. To optimize distance penalized objectives, we make use of the majorization-minimization principle. Resulting algorithms constructed within this framework are amenable to acceleration and come with global convergence guarantees. Applications to shape constraints, sparse regression, and rank-restricted matrix regression on synthetic and real data showcase the strong empirical performance of distance penalization, even under non-convex constraints.

Addressing the will to give a more complete picture than an average relationship provided by standard regression, a novel framework for estimating and predicting simultaneously several conditional quantiles is introduced. The proposed methodology leverages kernel-based multi-task learning to curb the embarrassing phenomenon of quantile crossing, with a one-step estimation procedure and no post-processing. Moreover, this framework comes along with theoretical guarantees and an efficient coordinate descent learning algorithm. Numerical experiments on benchmark and real datasets highlight the enhancements of our approach regarding the prediction error, the crossing occurrences and the training time.

Nystr\{o}m method has been used successfully to improve the computational efficiency of kernel ridge regression (KRR). Recently, theoretical analysis of Nystr\{o}m KRR, including generalization bound and convergence rate, has been established based on reproducing kernel Hilbert space (RKHS) associated with the symmetric positive semi-definite kernel. However, in real world applications, RKHS is not always optimal and kernel function is not necessary to be symmetric or positive semi-definite. In this paper, we consider the generalized Nystr\{o}m kernel regression (GNKR) with $\ell_2$ coefficient regularization, where the kernel just requires the continuity and boundedness. Error analysis is provided to characterize its generalization performance and the column norm sampling is introduced to construct the refined hypothesis space. In particular, the fast learning rate with polynomial decay is reached for the GNKR. Experimental analysis demonstrates the satisfactory performance of GNKR with the column norm sampling.

We study regression and classification in a setting where the learning algorithm is allowed to access only a limited number of attributes per example, known as the limited attribute observation model. In this well-studied model, we provide the first lower bounds giving a limit on the precision attainable by any algorithm for several variants of regression, notably linear regression with the absolute loss and the squared loss, as well as for classification with the hinge loss. We complement these lower bounds with a general purpose algorithm that gives an upper bound on the achievable precision limit in the setting of learning with missing data.

We introduce the framework of {\em blind regression} motivated by {\em matrix completion} for recommendation systems: given $m$ users, $n$ movies, and a subset of user-movie ratings, the goal is to predict the unobserved user-movie ratings given the data, i.e., to complete the partially observed matrix. Following the framework of non-parametric statistics, we posit that user $u$ and movie $i$ have features $x_1(u)$ and $x_2(i)$ respectively, and their corresponding rating $y(u,i)$ is a noisy measurement of $f(x_1(u), x_2(i))$ for some unknown function $f$. In contrast with classical regression, the features $x = (x_1(u), x_2(i))$ are not observed, making it challenging to apply standard regression methods to predict the unobserved ratings. Inspired by the classical Taylor's expansion for differentiable functions, we provide a prediction algorithm that is consistent for all Lipschitz functions. In fact, the analysis through our framework naturally leads to a variant of collaborative filtering, shedding insight into the widespread success of collaborative filtering in practice. Assuming each entry is sampled independently with probability at least $\max(m^{-1+\delta},n^{-1/2+\delta})$ with $\delta > 0$, we prove that the expected fraction of our estimates with error greater than $\epsilon$ is less than $\gamma^2 / \epsilon^2$ plus a polynomially decaying term, where $\gamma^2$ is the variance of the additive entry-wise noise term. Experiments with the MovieLens and Netflix datasets suggest that our algorithm provides principled improvements over basic collaborative filtering and is competitive with matrix factorization methods.

We propose a sparse and low-rank tensor regression model to relate a univariate outcome to a feature tensor, in which each unit-rank tensor from the CP decomposition of the coefficient tensor is assumed to be sparse. This structure is both parsimonious and highly interpretable, as it implies that the outcome is related to the features through a few distinct pathways, each of which may only involve subsets of feature dimensions. We take a divide-and-conquer strategy to simplify the task into a set of sparse unit-rank tensor regression problems. To make the computation efficient and scalable, for the unit-rank tensor regression, we propose a stagewise estimation procedure to efficiently trace out its entire solution path. We show that as the step size goes to zero, the stagewise solution paths converge exactly to those of the corresponding regularized regression. The superior performance of our approach is demonstrated on various real-world and synthetic examples.

Multi-response linear models aggregate a set of vanilla linear models by assuming correlated noise across them, which has an unknown covariance structure. To find the coefficient vector, estimators with a joint approximation of the noise covariance are often preferred than the simple linear regression in view of their superior empirical performance, which can be generally solved by alternating-minimization type procedures. Due to the non-convex nature of such joint estimators, the theoretical justification of their efficiency is typically challenging. The existing analyses fail to fully explain the empirical observations due to the assumption of resampling on the alternating procedures, which requires access to fresh samples in each iteration. In this work, we present a resampling-free analysis for the alternating minimization algorithm applied to the multi-response regression. In particular, we focus on the high-dimensional setting of multi-response linear models with structured coefficient parameter, and the statistical error of the parameter can be expressed by the complexity measure, Gaussian width, which is related to the assumed structure. More importantly, to the best of our knowledge, our result reveals for the first time that the alternating minimization with random initialization can achieve the same performance as the well-initialized one when solving this multi-response regression problem. Experimental results support our theoretical developments.

Neural Information Processing Systems http://nips.cc/

Data privacy is important in the AI era, and differential privacy (DP) is one of the golden solutions. However, DP is typically applicable only if data have a bounded underlying distribution. We address this limitation by leveraging second-moment information from a small amount of public data. We propose Public-moment-guided Truncation (PMT), which transforms private data using the public second-moment matrix and applies a principled truncation whose radius depends only on non-private quantities: data dimension and sample size. This transformation yields a well-conditioned second-moment matrix, enabling its inversion with a significantly strengthened ability to resist the DP noise. Furthermore, we demonstrate the applicability of PMT by using penalized and generalized linear regressions. Specifically, we design new loss functions and algorithms, ensuring that solutions in the transformed space can be mapped back to the original domain. We have established improvements in the models' DP estimation through theoretical error bounds, robustness guarantees, and convergence results, attributing the gains to the conditioning effect of PMT. Experiments on synthetic and real datasets confirm that PMT substantially improves the accuracy and stability of DP models.