In Learning with Label Proportions (LLP), the objective is to learn a supervised classifier when, instead of labels, only label proportions for bags of observations are known. This setting has broad practical relevance, in particular for privacy preserving data processing. We first show that the mean operator, a statistic which aggregates all labels, is minimally sufficient for the minimization of many proper scoring losses with linear (or kernelized) classifiers without using labels. We provide a fast learning algorithm that estimates the mean operator via a manifold regularizer with guaranteed approximation bounds. Then, we present an iterative learning algorithm that uses this as initialization. We ground this algorithm in Rademacher-style generalization bounds that fit the LLP setting, introducing a generalization of Rademacher complexity and a Label Proportion Complexity measure. This latter algorithm optimizes tractable bounds for the corresponding bag-empirical risk. Experiments are provided on fourteen domains, whose size ranges up to 300K observations. They display that our algorithms are scalable and tend to consistently outperform the state of the art in LLP. Moreover, in many cases, our algorithms compete with or are just percents of AUC away from the Oracle that learns knowing all labels. On the largest domains, half a dozen proportions can suffice, i.e. roughly 40K times less than the total number of labels.

We consider the problem of minimizing block-separable (non-smooth) convex functions subject to linear constraints. While the Alternating Direction Method of Multipliers (ADMM) for two-block linear constraints has been intensively studied both theoretically and empirically, in spite of some preliminary work, effective generalizations of ADMM to multiple blocks is still unclear. In this paper, we propose a parallel randomized block coordinate method named Parallel Direction Method of Multipliers (PDMM) to solve optimization problems with multi-block linear constraints. At each iteration, PDMM randomly updates some blocks in parallel, behaving like parallel randomized block coordinate descent. We establish the global convergence and the iteration complexity for PDMM with constant step size. We also show that PDMM can do randomized block coordinate descent on overlapping blocks. Experimental results show that PDMM performs better than state-of-the-arts methods in two applications, robust principal component analysis and overlapping group lasso.

Many machine learning approaches are characterized by information constraints on how they interact with the training data. These include memory and sequential access constraints (e.g. fast first-order methods to solve stochastic optimization problems); communication constraints (e.g. distributed learning); partial access to the underlying data (e.g. missing features and multi-armed bandits) and more. However, currently we have little understanding how such information constraints fundamentally affect our performance, independent of the learning problem semantics. For example, are there learning problems where any algorithm which has small memory footprint (or can use any bounded number of bits from each example, or has certain communication constraints) will perform worse than what is possible without such constraints? In this paper, we describe how a single set of results implies positive answers to the above, for several different settings.

Much of research in machine learning has centered around the search for inference algorithms that are both general-purpose and efficient. The problem is extremely challenging and general inference remains computationally expensive. We seek to address this problem by observing that in most specific applications of a model, we typically only need to perform a small subset of all possible inference computations. Motivated by this, we introduce just-in-time learning, a framework for fast and flexible inference that learns to speed up inference at run-time. Through a series of experiments, we show how this framework can allow us to combine the flexibility of sampling with the efficiency of deterministic message-passing.

Pairwise clustering methods partition the data space into clusters by the pairwise similarity between data points. The success of pairwise clustering largely depends on the pairwise similarity function defined over the data points, where kernel similarity is broadly used. In this paper, we present a novel pairwise clustering framework by bridging the gap between clustering and multi-class classification. This pairwise clustering framework learns an unsupervised nonparametric classifier from each data partition, and search for the optimal partition of the data by minimizing the generalization error of the learned classifiers associated with the data partitions. We consider two nonparametric classifiers in this framework, i.e. the nearest neighbor classifier and the plug-in classifier. Modeling the underlying data distribution by nonparametric kernel density estimation, the generalization error bounds for both unsupervised nonparametric classifiers are the sum of nonparametric pairwise similarity terms between the data points for the purpose of clustering. Under uniform distribution, the nonparametric similarity terms induced by both unsupervised classifiers exhibit a well known form of kernel similarity. We also prove that the generalization error bound for the unsupervised plug-in classifier is asymptotically equal to the weighted volume of cluster boundary for Low Density Separation, a widely used criteria for semi-supervised learning and clustering. Based on the derived nonparametric pairwise similarity using the plug-in classifier, we propose a new nonparametric exemplar-based clustering method with enhanced discriminative capability, whose superiority is evidenced by the experimental results.

We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response $y$, conditional on the model being selected (``condition on selection framework). This allows us to construct valid confidence intervals and hypothesis tests for regression coefficients that account for the selection procedure. In contrast to recent work in high-dimensional statistics, our results are exact (non-asymptotic) and require no eigenvalue-like assumptions on the design matrix $X$. Furthermore, the computational cost of marginal regression, constructing confidence intervals and hypothesis testing is negligible compared to the cost of linear regression, thus making our methods particularly suitable for extremely large datasets. Although we focus on marginal screening to illustrate the applicability of the condition on selection framework, this framework is much more broadly applicable. We show how to apply the proposed framework to several other selection procedures including orthogonal matching pursuit and marginal screening+Lasso."

We propose a new method named calibrated multivariate regression (CMR) for fitting high dimensional multivariate regression models. Compared to existing methods, CMR calibrates the regularization for each regression task with respect to its noise level so that it is simultaneously tuning insensitive and achieves an improved finite-sample performance. Computationally, we develop an efficient smoothed proximal gradient algorithm which has a worst-case iteration complexity $O(1/\epsilon)$, where $\epsilon$ is a pre-specified numerical accuracy. Theoretically, we prove that CMR achieves the optimal rate of convergence in parameter estimation. We illustrate the usefulness of CMR by thorough numerical simulations and show that CMR consistently outperforms other high dimensional multivariate regression methods. We also apply CMR on a brain activity prediction problem and find that CMR is as competitive as the handcrafted model created by human experts.

Canonical Correlation Analysis (CCA) is a widely used statistical tool with both well established theory and favorable performance for a wide range of machine learning problems. However, computing CCA for huge datasets can be very slow since it involves implementing QR decomposition or singular value decomposition of huge matrices. In this paper we introduce L-CCA, an iterative algorithm which can compute CCA fast on huge sparse datasets. Theory on both the asymptotic convergence and finite time accuracy of L-CCA are established. The experiments also show that L-CCA outperform other fast CCA approximation schemes on two real datasets.

We describe a framework for defining high-order image models that can be used in a variety of applications. The approach involves modeling local patterns in a multiscale representation of an image. Local properties of a coarsened image reflect non-local properties of the original image. In the case of binary images local properties are defined by the binary patterns observed over small neighborhoods around each pixel. With the multiscale representation we capture the frequency of patterns observed at different scales of resolution. This framework leads to expressive priors that depend on a relatively small number of parameters. For inference and learning we use an MCMC method for block sampling with very large blocks. We evaluate the approach with two example applications. One involves contour detection. The other involves binary segmentation.

Selecting a small informative subset from a given dataset, also called column sampling, has drawn much attention in machine learning. For incorporating structured data information into column sampling, research efforts were devoted to the cases where data points are fitted with clusters, simplices, or general convex hulls. This paper aims to study nonconvex hull learning which has rarely been investigated in the literature. In order to learn data-adaptive nonconvex hulls, a novel approach is proposed based on a graph-theoretic measure that leverages graph cycles to characterize the structural complexities of input data points. Employing this measure, we present a greedy algorithmic framework, dubbed Zeta Hulls, to perform structured column sampling. The process of pursuing a Zeta hull involves the computation of matrix inverse. To accelerate the matrix inversion computation and reduce its space complexity as well, we exploit a low-rank approximation to the graph adjacency matrix by using an efficient anchor graph technique. Extensive experimental results show that data representation learned by Zeta Hulls can achieve state-of-the-art accuracy in text and image classification tasks.