Nonconvex One-bit Single-label Multi-label Learning

We study an extreme scenario in multi-label learning where each training instance is endowed with a single one-bit label out of multiple labels. We formulate this problem as a non-trivial special case of one-bit rank-one matrix sensing and develop an efficient non-convex algorithm based on alternating power iteration. The proposed algorithm is able to recover the underlying low-rank matrix model with linear convergence. For a rank-$k$ model with $d_1$ features and $d_2$ classes, the proposed algorithm achieves $O(\epsilon)$ recovery error after retrieving $O(k^{1.5}d_1 d_2/\epsilon)$ one-bit labels within $O(kd)$ memory. Our bound is nearly optimal in the order of $O(1/\epsilon)$. This significantly improves the state-of-the-art sampling complexity of one-bit multi-label learning. We perform experiments to verify our theory and evaluate the performance of the proposed algorithm.

A Non-convex One-Pass Framework for Generalized Factorization Machine and Rank-One Matrix Sensing

We develop an efficient alternating framework for learning a generalized version of Factorization Machine (gFM) on steaming data with provable guarantees. When the instances are sampled from $d$ dimensional random Gaussian vectors and the target second order coefficient matrix in gFM is of rank $k$, our algorithm converges linearly, achieves $O(\epsilon)$ recovery error after retrieving $O(k^{3}d\log(1/\epsilon))$ training instances, consumes $O(kd)$ memory in one-pass of dataset and only requires matrix-vector product operations in each iteration. The key ingredient of our framework is a construction of an estimation sequence endowed with a so-called Conditionally Independent RIP condition (CI-RIP). As special cases of gFM, our framework can be applied to symmetric or asymmetric rank-one matrix sensing problems, such as inductive matrix completion and phase retrieval.

Large-scale Collaborative Imaging Genetics Studies of Risk Genetic Factors for Alzheimer's Disease Across Multiple Institutions

Genome-wide association studies (GWAS) offer new opportunities to identify genetic risk factors for Alzheimer's disease (AD). Recently, collaborative efforts across different institutions emerged that enhance the power of many existing techniques on individual institution data. However, a major barrier to collaborative studies of GWAS is that many institutions need to preserve individual data privacy. To address this challenge, we propose a novel distributed framework, termed Local Query Model (LQM) to detect risk SNPs for AD across multiple research institutions. To accelerate the learning process, we propose a Distributed Enhanced Dual Polytope Projection (D-EDPP) screening rule to identify irrelevant features and remove them from the optimization. To the best of our knowledge, this is the first successful run of the computationally intensive model selection procedure to learn a consistent model across different institutions without compromising their privacy while ranking the SNPs that may collectively affect AD. Empirical studies are conducted on 809 subjects with 5.9 million SNP features which are distributed across three individual institutions. D-EDPP achieved a 66-fold speed-up by effectively identifying irrelevant features.

Seeing the Forest from the Trees in Two Looks: Matrix Sketching by Cascaded Bilateral Sampling

Matrix sketching is aimed at finding close approximations of a matrix by factors of much smaller dimensions, which has important applications in optimization and machine learning. Given a matrix A of size m by n, state-of-the-art randomized algorithms take O(m * n) time and space to obtain its low-rank decomposition. Although quite useful, the need to store or manipulate the entire matrix makes it a computational bottleneck for truly large and dense inputs. Can we sketch an m-by-n matrix in O(m + n) cost by accessing only a small fraction of its rows and columns, without knowing anything about the remaining data? In this paper, we propose the cascaded bilateral sampling (CABS) framework to solve this problem. We start from demonstrating how the approximation quality of bilateral matrix sketching depends on the encoding powers of sampling. In particular, the sampled rows and columns should correspond to the code-vectors in the ground truth decompositions. Motivated by this analysis, we propose to first generate a pilot-sketch using simple random sampling, and then pursue more advanced, "follow-up" sampling on the pilot-sketch factors seeking maximal encoding powers. In this cascading process, the rise of approximation quality is shown to be lower-bounded by the improvement of encoding powers in the follow-up sampling step, thus theoretically guarantees the algorithmic boosting property. Computationally, our framework only takes linear time and space, and at the same time its performance rivals the quality of state-of-the-art algorithms consuming a quadratic amount of resources. Empirical evaluations on benchmark data fully demonstrate the potential of our methods in large scale matrix sketching and related areas.

Linear Convergence of Variance-Reduced Stochastic Gradient without Strong Convexity

Stochastic gradient algorithms estimate the gradient based on only one or a few samples and enjoy low computational cost per iteration. They have been widely used in large-scale optimization problems. However, stochastic gradient algorithms are usually slow to converge and achieve sub-linear convergence rates, due to the inherent variance in the gradient computation. To accelerate the convergence, some variance-reduced stochastic gradient algorithms, e.g., proximal stochastic variance-reduced gradient (Prox-SVRG) algorithm, have recently been proposed to solve strongly convex problems. Under the strongly convex condition, these variance-reduced stochastic gradient algorithms achieve a linear convergence rate. However, many machine learning problems are convex but not strongly convex. In this paper, we introduce Prox-SVRG and its projected variant called Variance-Reduced Projected Stochastic Gradient (VRPSG) to solve a class of non-strongly convex optimization problems widely used in machine learning. As the main technical contribution of this paper, we show that both VRPSG and Prox-SVRG achieve a linear convergence rate without strong convexity. A key ingredient in our proof is a Semi-Strongly Convex (SSC) inequality which is the first to be rigorously proved for a class of non-strongly convex problems in both constrained and regularized settings. Moreover, the SSC inequality is independent of algorithms and may be applied to analyze other stochastic gradient algorithms besides VRPSG and Prox-SVRG, which may be of independent interest. To the best of our knowledge, this is the first work that establishes the linear convergence rate for the variance-reduced stochastic gradient algorithms on solving both constrained and regularized problems without strong convexity.

Safe Screening for Multi-Task Feature Learning with Multiple Data Matrices

Multi-task feature learning (MTFL) is a powerful technique in boosting the predictive performance by learning multiple related classification/regression/clustering tasks simultaneously. However, solving the MTFL problem remains challenging when the feature dimension is extremely large. In this paper, we propose a novel screening rule---that is based on the dual projection onto convex sets (DPC)---to quickly identify the inactive features---that have zero coefficients in the solution vectors across all tasks. One of the appealing features of DPC is that: it is safe in the sense that the detected inactive features are guaranteed to have zero coefficients in the solution vectors across all tasks. Thus, by removing the inactive features from the training phase, we may have substantial savings in the computational cost and memory usage without sacrificing accuracy. To the best of our knowledge, it is the first screening rule that is applicable to sparse models with multiple data matrices. A key challenge in deriving DPC is to solve a nonconvex problem. We show that we can solve for the global optimum efficiently via a properly chosen parametrization of the constraint set. Moreover, DPC has very low computational cost and can be integrated with any existing solvers. We have evaluated the proposed DPC rule on both synthetic and real data sets. The experiments indicate that DPC is very effective in identifying the inactive features---especially for high dimensional data---which leads to a speedup up to several orders of magnitude.

Persistent Homology in Sparse Regression and Its Application to Brain Morphometry

Sparse systems are usually parameterized by a tuning parameter that determines the sparsity of the system. How to choose the right tuning parameter is a fundamental and difficult problem in learning the sparse system. In this paper, by treating the the tuning parameter as an additional dimension, persistent homological structures over the parameter space is introduced and explored. The structures are then further exploited in speeding up the computation using the proposed soft-thresholding technique. The topological structures are further used as multivariate features in the tensor-based morphometry (TBM) in characterizing white matter alterations in children who have experienced severe early life stress and maltreatment. These analyses reveal that stress-exposed children exhibit more diffuse anatomical organization across the whole white matter region.

Stochastic Coordinate Coding and Its Application for Drosophila Gene Expression Pattern Annotation

\textit{Drosophila melanogaster} has been established as a model organism for investigating the fundamental principles of developmental gene interactions. The gene expression patterns of \textit{Drosophila melanogaster} can be documented as digital images, which are annotated with anatomical ontology terms to facilitate pattern discovery and comparison. The automated annotation of gene expression pattern images has received increasing attention due to the recent expansion of the image database. The effectiveness of gene expression pattern annotation relies on the quality of feature representation. Previous studies have demonstrated that sparse coding is effective for extracting features from gene expression images. However, solving sparse coding remains a computationally challenging problem, especially when dealing with large-scale data sets and learning large size dictionaries. In this paper, we propose a novel algorithm to solve the sparse coding problem, called Stochastic Coordinate Coding (SCC). The proposed algorithm alternatively updates the sparse codes via just a few steps of coordinate descent and updates the dictionary via second order stochastic gradient descent. The computational cost is further reduced by focusing on the non-zero components of the sparse codes and the corresponding columns of the dictionary only in the updating procedure. Thus, the proposed algorithm significantly improves the efficiency and the scalability, making sparse coding applicable for large-scale data sets and large dictionary sizes. Our experiments on Drosophila gene expression data sets demonstrate the efficiency and the effectiveness of the proposed algorithm.

Lasso Screening Rules via Dual Polytope Projection

Lasso is a widely used regression technique to find sparse representations. When the dimension of the feature space and the number of samples are extremely large, solving the Lasso problem remains challenging. To improve the efficiency of solving large-scale Lasso problems, El Ghaoui and his colleagues have proposed the SAFE rules which are able to quickly identify the inactive predictors, i.e., predictors that have $0$ components in the solution vector. Then, the inactive predictors or features can be removed from the optimization problem to reduce its scale. By transforming the standard Lasso to its dual form, it can be shown that the inactive predictors include the set of inactive constraints on the optimal dual solution. In this paper, we propose an efficient and effective screening rule via Dual Polytope Projections (DPP), which is mainly based on the uniqueness and nonexpansiveness of the optimal dual solution due to the fact that the feasible set in the dual space is a convex and closed polytope. Moreover, we show that our screening rule can be extended to identify inactive groups in group Lasso. To the best of our knowledge, there is currently no "exact" screening rule for group Lasso. We have evaluated our screening rule using synthetic and real data sets. Results show that our rule is more effective in identifying inactive predictors than existing state-of-the-art screening rules for Lasso.

Two-Layer Feature Reduction for Sparse-Group Lasso via Decomposition of Convex Sets

Sparse-Group Lasso (SGL) has been shown to be a powerful regression technique for simultaneously discovering group and within-group sparse patterns by using a combination of the $\ell_1$ and $\ell_2$ norms. However, in large-scale applications, the complexity of the regularizers entails great computational challenges. In this paper, we propose a novel Two-Layer Feature REduction method (TLFre) for SGL via a decomposition of its dual feasible set. The two-layer reduction is able to quickly identify the inactive groups and the inactive features, respectively, which are guaranteed to be absent from the sparse representation and can be removed from the optimization. Existing feature reduction methods are only applicable for sparse models with one sparsity-inducing regularizer. To our best knowledge, TLFre is the first one that is capable of dealing with multiple sparsity-inducing regularizers. Moreover, TLFre has a very low computational cost and can be integrated with any existing solvers. We also develop a screening method---called DPC (DecomPosition of Convex set)---for the nonnegative Lasso problem. Experiments on both synthetic and real data sets show that TLFre and DPC improve the efficiency of SGL and nonnegative Lasso by several orders of magnitude.