Fitting statistical models is computationally challenging when the sample size or the dimension of the dataset is huge. An attractive approach for down-scaling the problem size is to first partition the dataset into subsets and then fit using distributed algorithms. The dataset can be partitioned either horizontally (in the sample space) or vertically (in the feature space). While the majority of the literature focuses on sample space partitioning, feature space partitioning is more effective when p n. Existing methods for partitioning features, however, are either vulnerable to high correlations or inefficient in reducing the model dimension. In this paper, we solve these problems through a new embarrassingly parallel framework named DECO for distributed variable selection and parameter estimation. In DECO, variables are first partitioned and allocated to m distributed workers. The decorrelated subset data within each worker are then fitted via any algorithm designed for high-dimensional problems. We show that by incorporating the decorrelation step, DECO can achieve consistent variable selection and parameter estimation on each subset with (almost) no assumptions. In addition, the convergence rate is nearly minimax optimal for both sparse and weakly sparse models and does NOT depend on the partition number m. Extensive numerical experiments are provided to illustrate the performance of the new framework.

This paper deals with price optimization, which is to find the best pricing strategy that maximizes revenue or profit, on the basis of demand forecasting models. Though recent advances in regression technologies have made it possible to reveal price-demand relationship of a large number of products, most existing price optimization methods, such as mixed integer programming formulation, cannot handle tens or hundreds of products because of their high computational costs. To cope with this problem, this paper proposes a novel approach based on network flow algorithms. We reveal a connection between supermodularity of the revenue and cross elasticity of demand. On the basis of this connection, we propose an efficient algorithm that employs network flow algorithms. The proposed algorithm can handle hundreds or thousands of products, and returns an exact optimal solution under an assumption regarding cross elasticity of demand. Even if the assumption does not hold, the proposed algorithm can efficiently find approximate solutions as good as other state-of-the-art methods, as empirical results show.

We develop methods for rapidly identifying important components of a convex optimization problem for the purpose of achieving fast convergence times.

We introduce a theoretical and algorithmic framework for multi-way graph partitioning that relies on a multiplicative cut-based objective. We refer to this objective as the Product Cut. We provide a detailed investigation of the mathematical properties of this objective and an effective algorithm for its optimization. The proposed model has strong mathematical underpinnings, and the corresponding algorithm achieves state-of-the-art performance on benchmark data sets.

Bregman divergences play a central role in the design and analysis of a range of machine learning algorithms through a handful of popular theorems. We present a new theorem which shows that "Bregman distortions" (employing a potentially non-convex generator) may be exactly re-written as a scaled Bregman divergence computed over transformed data. This property can be viewed from the standpoints of geometry (a scaled isometry with adaptive metrics) or convex optimization (relating generalized perspective transforms). Admissible distortions include geodesic distances on curved manifolds and projections or gauge-normalisation. Our theorem allows one to leverage to the wealth and convenience of Bregman divergences when analysing algorithms relying on the aforementioned Bregman distortions. We illustrate this with three novel applications of our theorem: a reduction from multi-class density ratio to class-probability estimation, a new adaptive projection free yet norm-enforcing dual norm mirror descent algorithm, and a reduction from clustering on flat manifolds to clustering on curved manifolds. Experiments on each of these domains validate the analyses and suggest that the scaled Bregman theorem might be a worthy addition to the popular handful of Bregman divergence properties that have been pervasive in machine learning.

One of the major issues in stochastic gradient descent (SGD) methods is how to choose an appropriate step size while running the algorithm. Since the traditional line search technique does not apply for stochastic optimization methods, the common practice in SGD is either to use a diminishing step size, or to tune a step size by hand, which can be time consuming in practice. In this paper, we propose to use the Barzilai-Borwein (BB) method to automatically compute step sizes for SGD and its variant: stochastic variance reduced gradient (SVRG) method, which leads to two algorithms: SGD-BB and SVRG-BB. We prove that SVRG-BB converges linearly for strongly convex objective functions. As a by-product, we prove the linear convergence result of SVRG with Option I proposed in [10], whose convergence result has been missing in the literature. Numerical experiments on standard data sets show that the performance of SGD-BB and SVRG-BB is comparable to and sometimes even better than SGD and SVRG with best-tuned step sizes, and is superior to some advanced SGD variants.

Stochastic gradient methods for machine learning and optimization problems are usually analyzed assuming data points are sampled with replacement. In contrast, sampling without replacement is far less understood, yet in practice it is very common, often easier to implement, and usually performs better. In this paper, we provide competitive convergence guarantees for without-replacement sampling under several scenarios, focusing on the natural regime of few passes over the data. Moreover, we describe a useful application of these results in the context of distributed optimization with randomly-partitioned data, yielding a nearly-optimal algorithm for regularized least squares (in terms of both communication complexity and runtime complexity) under broad parameter regimes. Our proof techniques combine ideas from stochastic optimization, adversarial online learning and transductive learning theory, and can potentially be applied to other stochastic optimization and learning problems.

We study k-SVD that is to obtain the first k singular vectors of a matrix A. Recently, a few breakthroughs have been discovered on k-SVD: Musco and Musco [19] proved the first gap-free convergence result using the block Krylov method, Shamir [21] discovered the first variance-reduction stochastic method, and Bhojanapalli et al. [7] provided the fastest O(nnz(A) + poly(1/ε))-time algorithm using alternating minimization. In this paper, we put forward a new and simple LazySVD framework to improve the above breakthroughs. This framework leads to a faster gap-free method outperforming [19], and the first accelerated and stochastic method outperforming [21]. In the O(nnz(A) + poly(1/ε)) running-time regime, LazySVD outperforms [7] in certain parameter regimes without even using alternating minimization.

In supervised binary hashing, one wants to learn a function that maps a highdimensional feature vector to a vector of binary codes, for application to fast image retrieval. This typically results in a difficult optimization problem, nonconvex and nonsmooth, because of the discrete variables involved. Much work has simply relaxed the problem during training, solving a continuous optimization, and truncating the codes a posteriori. This gives reasonable results but is quite suboptimal. Recent work has tried to optimize the objective directly over the binary codes and achieved better results, but the hash function was still learned a posteriori, which remains suboptimal. We propose a general framework for learning hash functions using affinity-based loss functions that uses auxiliary coordinates. This closes the loop and optimizes jointly over the hash functions and the binary codes so that they gradually match each other. The resulting algorithm can be seen as an iterated version of the procedure of optimizing first over the codes and then learning the hash function. Compared to this, our optimization is guaranteed to obtain better hash functions while being not much slower, as demonstrated experimentally in various supervised datasets.

Area under ROC (AUC) is a metric which is widely used for measuring the classification performance for imbalanced data. It is of theoretical and practical interest to develop online learning algorithms that maximizes AUC for large-scale data. A specific challenge in developing online AUC maximization algorithm is that the learning objective function is usually defined over a pair of training examples of opposite classes, and existing methods achieves on-line processing with higher space and time complexity. In this work, we propose a new stochastic online algorithm for AUC maximization. In particular, we show that AUC optimization can be equivalently formulated as a convex-concave saddle point problem. From this saddle representation, a stochastic online algorithm (SOLAM) is proposed which has time and space complexity of one datum. We establish theoretical convergence of SOLAM with high probability and demonstrate its effectiveness on standard benchmark datasets.