Learning Feature Nonlinearities with Non-Convex Regularized Binned Regression

For various applications, the relations between the dependent and independent variables are highly nonlinear. Consequently, for large scale complex problems, neural networks and regression trees are commonly preferred over linear models such as Lasso. This work proposes learning the feature nonlinearities by binning feature values and finding the best fit in each quantile using non-convex regularized linear regression. The algorithm first captures the dependence between neighboring quantiles by enforcing smoothness via piecewise-constant/linear approximation and then selects a sparse subset of good features. We prove that the proposed algorithm is statistically and computationally efficient. In particular, it achieves linear rate of convergence while requiring near-minimal number of samples. Evaluations on synthetic and real datasets demonstrate that algorithm is competitive with current state-of-the-art and accurately learns feature nonlinearities. Finally, we explore an interesting connection between the binning stage of our algorithm and sparse Johnson-Lindenstrauss matrices.

Sketching Meets Random Projection in the Dual: A Provable Recovery Algorithm for Big and High-dimensional Data

Sketching techniques have become popular for scaling up machine learning algorithms by reducing the sample size or dimensionality of massive data sets, while still maintaining the statistical power of big data. In this paper, we study sketching from an optimization point of view: we first show that the iterative Hessian sketch is an optimization process with preconditioning, and develop accelerated iterative Hessian sketch via the searching the conjugate direction; we then establish primal-dual connections between the Hessian sketch and dual random projection, and apply the preconditioned conjugate gradient approach on the dual problem, which leads to the accelerated iterative dual random projection methods. Finally to tackle the challenges from both large sample size and high-dimensionality, we propose the primal-dual sketch, which iteratively sketches the primal and dual formulations. We show that using a logarithmic number of calls to solvers of small scale problem, primal-dual sketch is able to recover the optimum of the original problem up to arbitrary precision. The proposed algorithms are validated via extensive experiments on synthetic and real data sets which complements our theoretical results.

Train and Test Tightness of LP Relaxations in Structured Prediction

Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation to the striking observation that approximations based on linear programming (LP) relaxations are often tight on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that tightness generalizes from train to test data.

Matrix Factorization with Explicit Trust and Distrust Relationships

With the advent of online social networks, recommender systems have became crucial for the success of many online applications/services due to their significance role in tailoring these applications to user-specific needs or preferences. Despite their increasing popularity, in general recommender systems suffer from the data sparsity and the cold-start problems. To alleviate these issues, in recent years there has been an upsurge of interest in exploiting social information such as trust relations among users along with the rating data to improve the performance of recommender systems. The main motivation for exploiting trust information in recommendation process stems from the observation that the ideas we are exposed to and the choices we make are significantly influenced by our social context. However, in large user communities, in addition to trust relations, the distrust relations also exist between users. For instance, in Epinions the concepts of personal "web of trust" and personal "block list" allow users to categorize their friends based on the quality of reviews into trusted and distrusted friends, respectively. In this paper, we propose a matrix factorization based model for recommendation in social rating networks that properly incorporates both trust and distrust relationships aiming to improve the quality of recommendations and mitigate the data sparsity and the cold-start users issues. Through experiments on the Epinions data set, we show that our new algorithm outperforms its standard trust-enhanced or distrust-enhanced counterparts with respect to accuracy, thereby demonstrating the positive effect that incorporation of explicit distrust information can have on recommender systems.

Exploiting Smoothness in Statistical Learning, Sequential Prediction, and Stochastic Optimization

In the last several years, the intimate connection between convex optimization and learning problems, in both statistical and sequential frameworks, has shifted the focus of algorithmic machine learning to examine this interplay. In particular, on one hand, this intertwinement brings forward new challenges in reassessment of the performance of learning algorithms including generalization and regret bounds under the assumptions imposed by convexity such as analytical properties of loss functions (e.g., Lipschitzness, strong convexity, and smoothness). On the other hand, emergence of datasets of an unprecedented size, demands the development of novel and more efficient optimization algorithms to tackle large-scale learning problems. The overarching goal of this thesis is to reassess the smoothness of loss functions in statistical learning, sequential prediction/online learning, and stochastic optimization and explicate its consequences. In particular we examine how smoothness of loss function could be beneficial or detrimental in these settings in terms of sample complexity, statistical consistency, regret analysis, and convergence rate, and investigate how smoothness can be leveraged to devise more efficient learning algorithms.

Recovering the Optimal Solution by Dual Random Projection

Random projection has been widely used in data classification. It maps high-dimensional data into a low-dimensional subspace in order to reduce the computational cost in solving the related optimization problem. While previous studies are focused on analyzing the classification performance of using random projection, in this work, we consider the recovery problem, i.e., how to accurately recover the optimal solution to the original optimization problem in the high-dimensional space based on the solution learned from the subspace spanned by random projections. We present a simple algorithm, termed Dual Random Projection, that uses the dual solution of the low-dimensional optimization problem to recover the optimal solution to the original problem. Our theoretical analysis shows that with a high probability, the proposed algorithm is able to accurately recover the optimal solution to the original problem, provided that the data matrix is of low rank or can be well approximated by a low rank matrix.

Excess Risk Bounds for Exponentially Concave Losses

The overarching goal of this paper is to derive excess risk bounds for learning from exp-concave loss functions in passive and sequential learning settings. Exp-concave loss functions encompass several fundamental problems in machine learning such as squared loss in linear regression, logistic loss in classification, and negative logarithm loss in portfolio management. In batch setting, we obtain sharp bounds on the performance of empirical risk minimization performed in a linear hypothesis space and with respect to the exp-concave loss functions. We also extend the results to the online setting where the learner receives the training examples in a sequential manner. We propose an online learning algorithm that is a properly modified version of online Newton method to obtain sharp risk bounds. Under an additional mild assumption on the loss function, we show that in both settings we are able to achieve an excess risk bound of $O(d\log n/n)$ that holds with a high probability.

Binary Excess Risk for Smooth Convex Surrogates

In statistical learning theory, convex surrogates of the 0-1 loss are highly preferred because of the computational and theoretical virtues that convexity brings in. This is of more importance if we consider smooth surrogates as witnessed by the fact that the smoothness is further beneficial both computationally- by attaining an {\it optimal} convergence rate for optimization, and in a statistical sense- by providing an improved {\it optimistic} rate for generalization bound. In this paper we investigate the smoothness property from the viewpoint of statistical consistency and show how it affects the binary excess risk. We show that in contrast to optimization and generalization errors that favor the choice of smooth surrogate loss, the smoothness of loss function may degrade the binary excess risk. Motivated by this negative result, we provide a unified analysis that integrates optimization error, generalization bound, and the error in translating convex excess risk into a binary excess risk when examining the impact of smoothness on the binary excess risk. We show that under favorable conditions appropriate choice of smooth convex loss will result in a binary excess risk that is better than $O(1/\sqrt{n})$.

Beating the Minimax Rate of Active Learning with Prior Knowledge

Active learning refers to the learning protocol where the learner is allowed to choose a subset of instances for labeling. Previous studies have shown that, compared with passive learning, active learning is able to reduce the label complexity exponentially if the data are linearly separable or satisfy the Tsybakov noise condition with parameter $\kappa=1$. In this paper, we propose a novel active learning algorithm using a convex surrogate loss, with the goal to broaden the cases for which active learning achieves an exponential improvement. We make use of a convex loss not only because it reduces the computational cost, but more importantly because it leads to a tight bound for the empirical process (i.e., the difference between the empirical estimation and the expectation) when the current solution is close to the optimal one. Under the assumption that the norm of the optimal classifier that minimizes the convex risk is available, our analysis shows that the introduction of the convex surrogate loss yields an exponential reduction in the label complexity even when the parameter $\kappa$ of the Tsybakov noise is larger than $1$. To the best of our knowledge, this is the first work that improves the minimax rate of active learning by utilizing certain priori knowledge.

MixedGrad: An O(1/T) Convergence Rate Algorithm for Stochastic Smooth Optimization

It is well known that the optimal convergence rate for stochastic optimization of smooth functions is $O(1/\sqrt{T})$, which is same as stochastic optimization of Lipschitz continuous convex functions. This is in contrast to optimizing smooth functions using full gradients, which yields a convergence rate of $O(1/T^2)$. In this work, we consider a new setup for optimizing smooth functions, termed as {\bf Mixed Optimization}, which allows to access both a stochastic oracle and a full gradient oracle. Our goal is to significantly improve the convergence rate of stochastic optimization of smooth functions by having an additional small number of accesses to the full gradient oracle. We show that, with an $O(\ln T)$ calls to the full gradient oracle and an $O(T)$ calls to the stochastic oracle, the proposed mixed optimization algorithm is able to achieve an optimization error of $O(1/T)$.