Preference Completion: Large-scale Collaborative Ranking from Pairwise Comparisons

In this paper we consider the collaborative ranking setting: a pool of users each provides a small number of pairwise preferences between $d$ possible items; from these we need to predict preferences of the users for items they have not yet seen. We do so by fitting a rank $r$ score matrix to the pairwise data, and provide two main contributions: (a) we show that an algorithm based on convex optimization provides good generalization guarantees once each user provides as few as $O(r\log^2 d)$ pairwise comparisons -- essentially matching the sample complexity required in the related matrix completion setting (which uses actual numerical as opposed to pairwise information), and (b) we develop a large-scale non-convex implementation, which we call AltSVM, that trains a factored form of the matrix via alternating minimization (which we show reduces to alternating SVM problems), and scales and parallelizes very well to large problem settings. It also outperforms common baselines on many moderately large popular collaborative filtering datasets in both NDCG and in other measures of ranking performance.

Optimal Decision-Theoretic Classification Using Non-Decomposable Performance Metrics

We provide a general theoretical analysis of expected out-of-sample utility, also referred to as decision-theoretic classification, for non-decomposable binary classification metrics such as F-measure and Jaccard coefficient. Our key result is that the expected out-of-sample utility for many performance metrics is provably optimized by a classifier which is equivalent to a signed thresholding of the conditional probability of the positive class. Our analysis bridges a gap in the literature on binary classification, revealed in light of recent results for non-decomposable metrics in population utility maximization style classification. Our results identify checkable properties of a performance metric which are sufficient to guarantee a probability ranking principle. We propose consistent estimators for optimal expected out-of-sample classification. As a consequence of the probability ranking principle, computational requirements can be reduced from exponential to cubic complexity in the general case, and further reduced to quadratic complexity in special cases. We provide empirical results on simulated and benchmark datasets evaluating the performance of the proposed algorithms for decision-theoretic classification and comparing them to baseline and state-of-the-art methods in population utility maximization for non-decomposable metrics.

PASSCoDe: Parallel ASynchronous Stochastic dual Co-ordinate Descent

Stochastic Dual Coordinate Descent (SDCD) has become one of the most efficient ways to solve the family of $\ell_2$-regularized empirical risk minimization problems, including linear SVM, logistic regression, and many others. The vanilla implementation of DCD is quite slow; however, by maintaining primal variables while updating dual variables, the time complexity of SDCD can be significantly reduced. Such a strategy forms the core algorithm in the widely-used LIBLINEAR package. In this paper, we parallelize the SDCD algorithms in LIBLINEAR. In recent research, several synchronized parallel SDCD algorithms have been proposed, however, they fail to achieve good speedup in the shared memory multi-core setting. In this paper, we propose a family of asynchronous stochastic dual coordinate descent algorithms (ASDCD). Each thread repeatedly selects a random dual variable and conducts coordinate updates using the primal variables that are stored in the shared memory. We analyze the convergence properties when different locking/atomic mechanisms are applied. For implementation with atomic operations, we show linear convergence under mild conditions. For implementation without any atomic operations or locking, we present the first {\it backward error analysis} for ASDCD under the multi-core environment, showing that the converged solution is the exact solution for a primal problem with perturbed regularizer. Experimental results show that our methods are much faster than previous parallel coordinate descent solvers.

Proximal Quasi-Newton for Computationally Intensive L1-regularized M-estimators

We consider the class of optimization problems arising from computationally intensive L1-regularized M-estimators, where the function or gradient values are very expensive to compute. A particular instance of interest is the L1-regularized MLE for learning Conditional Random Fields (CRFs), which are a popular class of statistical models for varied structured prediction problems such as sequence labeling, alignment, and classification with label taxonomy. L1-regularized MLEs for CRFs are particularly expensive to optimize since computing the gradient values requires an expensive inference step. In this work, we propose the use of a carefully constructed proximal quasi-Newton algorithm for such computationally intensive M-estimation problems, where we employ an aggressive active set selection technique. In a key contribution of the paper, we show that the proximal quasi-Newton method is provably super-linearly convergent, even in the absence of strong convexity, by leveraging a restricted variant of strong convexity. In our experiments, the proposed algorithm converges considerably faster than current state-of-the-art on the problems of sequence labeling and hierarchical classification.

A Scalable Asynchronous Distributed Algorithm for Topic Modeling

Learning meaningful topic models with massive document collections which contain millions of documents and billions of tokens is challenging because of two reasons: First, one needs to deal with a large number of topics (typically in the order of thousands). Second, one needs a scalable and efficient way of distributing the computation across multiple machines. In this paper we present a novel algorithm F+Nomad LDA which simultaneously tackles both these problems. In order to handle large number of topics we use an appropriately modified Fenwick tree. This data structure allows us to sample from a multinomial distribution over $T$ items in $O(\log T)$ time. Moreover, when topic counts change the data structure can be updated in $O(\log T)$ time. In order to distribute the computation across multiple processor we present a novel asynchronous framework inspired by the Nomad algorithm of \cite{YunYuHsietal13}. We show that F+Nomad LDA significantly outperform state-of-the-art on massive problems which involve millions of documents, billions of words, and thousands of topics.

PU Learning for Matrix Completion

In this paper, we consider the matrix completion problem when the observations are one-bit measurements of some underlying matrix M, and in particular the observed samples consist only of ones and no zeros. This problem is motivated by modern applications such as recommender systems and social networks where only "likes" or "friendships" are observed. The problem of learning from only positive and unlabeled examples, called PU (positive-unlabeled) learning, has been studied in the context of binary classification. We consider the PU matrix completion problem, where an underlying real-valued matrix M is first quantized to generate one-bit observations and then a subset of positive entries is revealed. Under the assumption that M has bounded nuclear norm, we provide recovery guarantees for two different observation models: 1) M parameterizes a distribution that generates a binary matrix, 2) M is thresholded to obtain a binary matrix. For the first case, we propose a "shifted matrix completion" method that recovers M using only a subset of indices corresponding to ones, while for the second case, we propose a "biased matrix completion" method that recovers the (thresholded) binary matrix. Both methods yield strong error bounds --- if M is n by n, the Frobenius error is bounded as O(1/((1-rho)n), where 1-rho denotes the fraction of ones observed. This implies a sample complexity of O(n\log n) ones to achieve a small error, when M is dense and n is large. We extend our methods and guarantees to the inductive matrix completion problem, where rows and columns of M have associated features. We provide efficient and scalable optimization procedures for both the methods and demonstrate the effectiveness of the proposed methods for link prediction (on real-world networks consisting of over 2 million nodes and 90 million links) and semi-supervised clustering tasks.

Large-scale Multi-label Learning with Missing Labels

The multi-label classification problem has generated significant interest in recent years. However, existing approaches do not adequately address two key challenges: (a) the ability to tackle problems with a large number (say millions) of labels, and (b) the ability to handle data with missing labels. In this paper, we directly address both these problems by studying the multi-label problem in a generic empirical risk minimization (ERM) framework. Our framework, despite being simple, is surprisingly able to encompass several recent label-compression based methods which can be derived as special cases of our method. To optimize the ERM problem, we develop techniques that exploit the structure of specific loss functions - such as the squared loss function - to offer efficient algorithms. We further show that our learning framework admits formal excess risk bounds even in the presence of missing labels. Our risk bounds are tight and demonstrate better generalization performance for low-rank promoting trace-norm regularization when compared to (rank insensitive) Frobenius norm regularization. Finally, we present extensive empirical results on a variety of benchmark datasets and show that our methods perform significantly better than existing label compression based methods and can scale up to very large datasets such as the Wikipedia dataset.

A Divide-and-Conquer Solver for Kernel Support Vector Machines

The kernel support vector machine (SVM) is one of the most widely used classification methods; however, the amount of computation required becomes the bottleneck when facing millions of samples. In this paper, we propose and analyze a novel divide-and-conquer solver for kernel SVMs (DC-SVM). In the division step, we partition the kernel SVM problem into smaller subproblems by clustering the data, so that each subproblem can be solved independently and efficiently. We show theoretically that the support vectors identified by the subproblem solution are likely to be support vectors of the entire kernel SVM problem, provided that the problem is partitioned appropriately by kernel clustering. In the conquer step, the local solutions from the subproblems are used to initialize a global coordinate descent solver, which converges quickly as suggested by our analysis. By extending this idea, we develop a multilevel Divide-and-Conquer SVM algorithm with adaptive clustering and early prediction strategy, which outperforms state-of-the-art methods in terms of training speed, testing accuracy, and memory usage. As an example, on the covtype dataset with half-a-million samples, DC-SVM is 7 times faster than LIBSVM in obtaining the exact SVM solution (to within $10^{-6}$ relative error) which achieves 96.15% prediction accuracy. Moreover, with our proposed early prediction strategy, DC-SVM achieves about 96% accuracy in only 12 minutes, which is more than 100 times faster than LIBSVM.

Sparse Inverse Covariance Matrix Estimation Using Quadratic Approximation

The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to recent state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modifications that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and present experimental results using synthetic and real-world application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods.

Provable Inductive Matrix Completion

Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matrix completion in the exact recovery setting. That is, we assume that the ratings matrix is generated by applying feature vectors to a low-rank matrix and the goal is to recover back the underlying matrix. Furthermore, we generalize the problem to that of low-rank matrix estimation using rank-1 measurements. We study this generic problem and provide conditions that the set of measurements should satisfy so that the alternating minimization method (which otherwise is a non-convex method with no convergence guarantees) is able to recover back the {\em exact} underlying low-rank matrix. In addition to inductive matrix completion, we show that two other low-rank estimation problems can be studied in our framework: a) general low-rank matrix sensing using rank-1 measurements, and b) multi-label regression with missing labels. For both the problems, we provide novel and interesting bounds on the number of measurements required by alternating minimization to provably converges to the {\em exact} low-rank matrix. In particular, our analysis for the general low rank matrix sensing problem significantly improves the required storage and computational cost than that required by the RIP-based matrix sensing methods \cite{RechtFP2007}. Finally, we provide empirical validation of our approach and demonstrate that alternating minimization is able to recover the true matrix for the above mentioned problems using a small number of measurements.