arXiv subjects

Qihang Lin

Publications and source records attributed to Qihang Lin.

At least 55 records · Page 3Linked to original sources

arXiv2018

RSG: Beating Subgradient Method without Smoothness and Strong Convexity

In this paper, we study the efficiency of a {\bf R}estarted {\bf S}ub{\bf G}radient (RSG) method that periodically restarts the standard subgradient method (SG). We show that, when applied to a broad class of convex optimization problems, RSG method can find an $ε$-optimal solution with a lower complexity than the SG method. In particular, we first show that RSG can reduce the dependence of SG's iteration complexity on the distance between the initial solution and the optimal set to that between the $ε$-level set and the optimal set {multiplied by a logarithmic factor}. Moreover, we show the advantages of RSG over SG in solving three different families of convex optimization problems. (a) For the problems whose epigraph is a polyhedron, RSG is shown to converge linearly. (b) For the problems with local quadratic growth property in the $ε$-sublevel set, RSG has an $O(\frac{1}ε\log(\frac{1}ε))$ iteration complexity. (c) For the problems that admit a local Kurdyka-Łojasiewicz property with a power constant of $β\in[0,1)$, RSG has an $O(\frac{1}{ε^{2β}}\log(\frac{1}ε))$ iteration complexity. The novelty of our analysis lies at exploiting the lower bound of the first-order optimality residual at the $ε$-level set. It is this novelty that allows us to explore the local properties of functions (e.g., local quadratic growth property, local Kurdyka-Łojasiewicz property, more generally local error bound conditions) to develop the improved convergence of RSG. { We also develop a practical variant of RSG enjoying faster convergence than the SG method, which can be run without knowing the involved parameters in the local error bound condition.} We demonstrate the effectiveness of the proposed algorithms on several machine learning tasks including regression, classification and matrix completion.

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arXiv2018

On Degrees of Freedom of Projection Estimators with Applications to Multivariate Nonparametric Regression

In this paper, we consider the nonparametric regression problem with multivariate predictors. We provide a characterization of the degrees of freedom and divergence for estimators of the unknown regression function, which are obtained as outputs of linearly constrained quadratic optimization procedures, namely, minimizers of the least squares criterion with linear constraints and/or quadratic penalties. As special cases of our results, we derive explicit expressions for the degrees of freedom in many nonparametric regression problems, e.g., bounded isotonic regression, multivariate (penalized) convex regression, and additive total variation regularization. Our theory also yields, as special cases, known results on the degrees of freedom of many well-studied estimators in the statistics literature, such as ridge regression, Lasso and generalized Lasso. Our results can be readily used to choose the tuning parameter(s) involved in the estimation procedure by minimizing the Stein's unbiased risk estimate. As a by-product of our analysis we derive an interesting connection between bounded isotonic regression and isotonic regression on a general partially ordered set, which is of independent interest.

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arXiv2018

A Unified Analysis of Stochastic Momentum Methods for Deep Learning

Stochastic momentum methods have been widely adopted in training deep neural networks. However, their theoretical analysis of convergence of the training objective and the generalization error for prediction is still under-explored. This paper aims to bridge the gap between practice and theory by analyzing the stochastic gradient (SG) method, and the stochastic momentum methods including two famous variants, i.e., the stochastic heavy-ball (SHB) method and the stochastic variant of Nesterov's accelerated gradient (SNAG) method. We propose a framework that unifies the three variants. We then derive the convergence rates of the norm of gradient for the non-convex optimization problem, and analyze the generalization performance through the uniform stability approach. Particularly, the convergence analysis of the training objective exhibits that SHB and SNAG have no advantage over SG. However, the stability analysis shows that the momentum term can improve the stability of the learned model and hence improve the generalization performance. These theoretical insights verify the common wisdom and are also corroborated by our empirical analysis on deep learning.

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arXiv2018

Block-Normalized Gradient Method: An Empirical Study for Training Deep Neural Network

In this paper, we propose a generic and simple strategy for utilizing stochastic gradient information in optimization. The technique essentially contains two consecutive steps in each iteration: 1) computing and normalizing each block (layer) of the mini-batch stochastic gradient; 2) selecting appropriate step size to update the decision variable (parameter) towards the negative of the block-normalized gradient. We conduct extensive empirical studies on various non-convex neural network optimization problems, including multi-layer perceptron, convolution neural networks and recurrent neural networks. The results indicate the block-normalized gradient can help accelerate the training of neural networks. In particular, we observe that the normalized gradient methods having constant step size with occasionally decay, such as SGD with momentum, have better performance in the deep convolution neural networks, while those with adaptive step sizes, such as Adam, perform better in recurrent neural networks. Besides, we also observe this line of methods can lead to solutions with better generalization properties, which is confirmed by the performance improvement over strong baselines.

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arXiv2018

Prophit: Causal inverse classification for multiple continuously valued treatment policies

Inverse classification uses an induced classifier as a queryable oracle to guide test instances towards a preferred posterior class label. The result produced from the process is a set of instance-specific feature perturbations, or recommendations, that optimally improve the probability of the class label. In this work, we adopt a causal approach to inverse classification, eliciting treatment policies (i.e., feature perturbations) for models induced with causal properties. In so doing, we solve a long-standing problem of eliciting multiple, continuously valued treatment policies, using an updated framework and corresponding set of assumptions, which we term the inverse classification potential outcomes framework (ICPOF), along with a new measure, referred to as the individual future estimated effects ($i$FEE). We also develop the approximate propensity score (APS), based on Gaussian processes, to weight treatments, much like the inverse propensity score weighting used in past works. We demonstrate the viability of our methods on student performance.

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arXiv2017

DSCOVR: Randomized Primal-Dual Block Coordinate Algorithms for Asynchronous Distributed Optimization

Machine learning with big data often involves large optimization models. For distributed optimization over a cluster of machines, frequent communication and synchronization of all model parameters (optimization variables) can be very costly. A promising solution is to use parameter servers to store different subsets of the model parameters, and update them asynchronously at different machines using local datasets. In this paper, we focus on distributed optimization of large linear models with convex loss functions, and propose a family of randomized primal-dual block coordinate algorithms that are especially suitable for asynchronous distributed implementation with parameter servers. In particular, we work with the saddle-point formulation of such problems which allows simultaneous data and model partitioning, and exploit its structure by doubly stochastic coordinate optimization with variance reduction (DSCOVR). Compared with other first-order distributed algorithms, we show that DSCOVR may require less amount of overall computation and communication, and less or no synchronization. We discuss the implementation details of the DSCOVR algorithms, and present numerical experiments on an industrial distributed computing system.

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arXiv2017

A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates

This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a linear optimization under the inequality constraint are time-consuming, which render both projected gradient methods and conditional gradient methods (a.k.a. the Frank-Wolfe algorithm) expensive. In this paper, we develop projection reduced optimization algorithms for both smooth and non-smooth optimization with improved convergence rates under a certain regularity condition of the constraint function. We first present a general theory of optimization with only one projection. Its application to smooth optimization with only one projection yields $O(1/ε)$ iteration complexity, which improves over the $O(1/ε^2)$ iteration complexity established before for non-smooth optimization and can be further reduced under strong convexity. Then we introduce a local error bound condition and develop faster algorithms for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we achieve an iteration complexity of $\widetilde O(1/ε^{2(1-θ)})$ for non-smooth optimization and $\widetilde O(1/ε^{1-θ})$ for smooth optimization, where $θ\in(0,1]$ appearing the local error bound condition characterizes the functional local growth rate around the optimal solutions. Novel applications in solving the constrained $\ell_1$ minimization problem and a positive semi-definite constrained distance metric learning problem demonstrate that the proposed algorithms achieve significant speed-up compared with previous algorithms.

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arXiv2017

A budget-constrained inverse classification framework for smooth classifiers

Inverse classification is the process of manipulating an instance such that it is more likely to conform to a specific class. Past methods that address such a problem have shortcomings. Greedy methods make changes that are overly radical, often relying on data that is strictly discrete. Other methods rely on certain data points, the presence of which cannot be guaranteed. In this paper we propose a general framework and method that overcomes these and other limitations. The formulation of our method can use any differentiable classification function. We demonstrate the method by using logistic regression and Gaussian kernel SVMs. We constrain the inverse classification to occur on features that can actually be changed, each of which incurs an individual cost. We further subject such changes to fall within a certain level of cumulative change (budget). Our framework can also accommodate the estimation of (indirectly changeable) features whose values change as a consequence of actions taken. Furthermore, we propose two methods for specifying feature-value ranges that result in different algorithmic behavior. We apply our method, and a proposed sensitivity analysis-based benchmark method, to two freely available datasets: Student Performance from the UCI Machine Learning Repository and a real world cardiovascular disease dataset. The results obtained demonstrate the validity and benefits of our framework and method.

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arXiv2017

Doubly Stochastic Primal-Dual Coordinate Method for Bilinear Saddle-Point Problem

We propose a doubly stochastic primal-dual coordinate optimization algorithm for empirical risk minimization, which can be formulated as a bilinear saddle-point problem. In each iteration, our method randomly samples a block of coordinates of the primal and dual solutions to update. The linear convergence of our method could be established in terms of 1) the distance from the current iterate to the optimal solution and 2) the primal-dual objective gap. We show that the proposed method has a lower overall complexity than existing coordinate methods when either the data matrix has a factorized structure or the proximal mapping on each block is computationally expensive, e.g., involving an eigenvalue decomposition. The efficiency of the proposed method is confirmed by empirical studies on several real applications, such as the multi-task large margin nearest neighbor problem.

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arXiv2017

Generalized Inverse Classification

Inverse classification is the process of perturbing an instance in a meaningful way such that it is more likely to conform to a specific class. Historical methods that address such a problem are often framed to leverage only a single classifier, or specific set of classifiers. These works are often accompanied by naive assumptions. In this work we propose generalized inverse classification (GIC), which avoids restricting the classification model that can be used. We incorporate this formulation into a refined framework in which GIC takes place. Under this framework, GIC operates on features that are immediately actionable. Each change incurs an individual cost, either linear or non-linear. Such changes are subjected to occur within a specified level of cumulative change (budget). Furthermore, our framework incorporates the estimation of features that change as a consequence of direct actions taken (indirectly changeable features). To solve such a problem, we propose three real-valued heuristic-based methods and two sensitivity analysis-based comparison methods, each of which is evaluated on two freely available real-world datasets. Our results demonstrate the validity and benefits of our formulation, framework, and methods.

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arXiv2016

Bayesian Decision Process for Cost-Efficient Dynamic Ranking via Crowdsourcing

Rank aggregation based on pairwise comparisons over a set of items has a wide range of applications. Although considerable research has been devoted to the development of rank aggregation algorithms, one basic question is how to efficiently collect a large amount of high-quality pairwise comparisons for the ranking purpose. Because of the advent of many crowdsourcing services, a crowd of workers are often hired to conduct pairwise comparisons with a small monetary reward for each pair they compare. Since different workers have different levels of reliability and different pairs have different levels of ambiguity, it is desirable to wisely allocate the limited budget for comparisons among the pairs of items and workers so that the global ranking can be accurately inferred from the comparison results. To this end, we model the active sampling problem in crowdsourced ranking as a Bayesian Markov decision process, which dynamically selects item pairs and workers to improve the ranking accuracy under a budget constraint. We further develop a computationally efficient sampling policy based on knowledge gradient as well as a moment matching technique for posterior approximation. Experimental evaluations on both synthetic and real data show that the proposed policy achieves high ranking accuracy with a lower labeling cost.

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arXiv2016

Homotopy Smoothing for Non-Smooth Problems with Lower Complexity than $O(1/ε)$

In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving such non-smooth optimization problems is $O(1/ε)$ without any assumption on the strong convexity. In this work, we will show that the proposed HOPS achieved a lower iteration complexity of $\widetilde O(1/ε^{1-θ})$\footnote{$\widetilde O()$ suppresses a logarithmic factor.} with $θ\in(0,1]$ capturing the local sharpness of the objective function around the optimal solutions. To the best of our knowledge, this is the lowest iteration complexity achieved so far for the considered non-smooth optimization problems without strong convexity assumption. The HOPS algorithm employs Nesterov's smoothing technique and Nesterov's accelerated gradient method and runs in stages, which gradually decreases the smoothing parameter in a stage-wise manner until it yields a sufficiently good approximation of the original function. We show that HOPS enjoys a linear convergence for many well-known non-smooth problems (e.g., empirical risk minimization with a piece-wise linear loss function and $\ell_1$ norm regularizer, finding a point in a polyhedron, cone programming, etc). Experimental results verify the effectiveness of HOPS in comparison with Nesterov's smoothing algorithm and the primal-dual style of first-order methods.

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arXiv2016

Optimal Stochastic Strongly Convex Optimization with a Logarithmic Number of Projections

We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent~(SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent~(Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Given a strongly convex optimization and for a total number of $T$ iterations, Epro-SGD requires only $\log(T)$ projections, and meanwhile attains an optimal convergence rate of $O(1/T)$, both in expectation and with a high probability. To exploit the structure of the optimization problem, we propose a proximal variant of Epro-SGD, namely Epro-ORDA, based on the optimal regularized dual averaging method. We apply the proposed methods on real-world applications; the empirical results demonstrate the effectiveness of our methods.

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arXiv2016

Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimization

Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of two stochastic momentum methods, namely stochastic heavy-ball method and the stochastic variant of Nesterov's accelerated gradient method. We hope that the basic convergence results developed in this paper can serve the reference to the convergence of stochastic momentum methods and also serve the baselines for comparison in future development of stochastic momentum methods. The novelty of convergence analysis presented in this paper is a unified framework, revealing more insights about the similarities and differences between different stochastic momentum methods and stochastic gradient method. The unified framework exhibits a continuous change from the gradient method to Nesterov's accelerated gradient method and finally the heavy-ball method incurred by a free parameter, which can help explain a similar change observed in the testing error convergence behavior for deep learning. Furthermore, our empirical results for optimizing deep neural networks demonstrate that the stochastic variant of Nesterov's accelerated gradient method achieves a good tradeoff (between speed of convergence in training error and robustness of convergence in testing error) among the three stochastic methods.

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arXiv2016

Stochastic subGradient Methods with Linear Convergence for Polyhedral Convex Optimization

In this paper, we show that simple {Stochastic} subGradient Decent methods with multiple Restarting, named {\bf RSGD}, can achieve a \textit{linear convergence rate} for a class of non-smooth and non-strongly convex optimization problems where the epigraph of the objective function is a polyhedron, to which we refer as {\bf polyhedral convex optimization}. Its applications in machine learning include $\ell_1$ constrained or regularized piecewise linear loss minimization and submodular function minimization. To the best of our knowledge, this is the first result on the linear convergence rate of stochastic subgradient methods for non-smooth and non-strongly convex optimization problems.

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arXiv2016

Distributed Stochastic Variance Reduced Gradient Methods and A Lower Bound for Communication Complexity

We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms.

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arXiv2015

On Data Preconditioning for Regularized Loss Minimization

In this work, we study data preconditioning, a well-known and long-existing technique, for boosting the convergence of first-order methods for regularized loss minimization. It is well understood that the condition number of the problem, i.e., the ratio of the Lipschitz constant to the strong convexity modulus, has a harsh effect on the convergence of the first-order optimization methods. Therefore, minimizing a small regularized loss for achieving good generalization performance, yielding an ill conditioned problem, becomes the bottleneck for big data problems. We provide a theory on data preconditioning for regularized loss minimization. In particular, our analysis exhibits an appropriate data preconditioner and characterizes the conditions on the loss function and on the data under which data preconditioning can reduce the condition number and therefore boost the convergence for minimizing the regularized loss. To make the data preconditioning practically useful, we endeavor to employ and analyze a random sampling approach to efficiently compute the preconditioned data. The preliminary experiments validate our theory.

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arXiv2015

Fast Sparse Least-Squares Regression with Non-Asymptotic Guarantees

In this paper, we study a fast approximation method for {\it large-scale high-dimensional} sparse least-squares regression problem by exploiting the Johnson-Lindenstrauss (JL) transforms, which embed a set of high-dimensional vectors into a low-dimensional space. In particular, we propose to apply the JL transforms to the data matrix and the target vector and then to solve a sparse least-squares problem on the compressed data with a {\it slightly larger regularization parameter}. Theoretically, we establish the optimization error bound of the learned model for two different sparsity-inducing regularizers, i.e., the elastic net and the $\ell_1$ norm. Compared with previous relevant work, our analysis is {\it non-asymptotic and exhibits more insights} on the bound, the sample complexity and the regularization. As an illustration, we also provide an error bound of the {\it Dantzig selector} under JL transforms.

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