arXiv subjects

Rong Jin

Publications and source records attributed to Rong Jin.

At least 145 records · Page 8Linked to original sources

arXiv2018

NEON+: Accelerated Gradient Methods for Extracting Negative Curvature for Non-Convex Optimization

Accelerated gradient (AG) methods are breakthroughs in convex optimization, improving the convergence rate of the gradient descent method for optimization with smooth functions. However, the analysis of AG methods for non-convex optimization is still limited. It remains an open question whether AG methods from convex optimization can accelerate the convergence of the gradient descent method for finding local minimum of non-convex optimization problems. This paper provides an affirmative answer to this question. In particular, we analyze two renowned variants of AG methods (namely Polyak's Heavy Ball method and Nesterov's Accelerated Gradient method) for extracting the negative curvature from random noise, which is central to escaping from saddle points. By leveraging the proposed AG methods for extracting the negative curvature, we present a new AG algorithm with double loops for non-convex optimization~\footnote{this is in contrast to a single-loop AG algorithm proposed in a recent manuscript~\citep{AGNON}, which directly analyzed the Nesterov's AG method for non-convex optimization and appeared online on November 29, 2017. However, we emphasize that our work is an independent work, which is inspired by our earlier work~\citep{NEON17} and is based on a different novel analysis.}, which converges to second-order stationary point $\x$ such that $\|\nabla f(\x)\|\leq ε$ and $\nabla^2 f(\x)\geq -\sqrtε I$ with $\widetilde O(1/ε^{1.75})$ iteration complexity, improving that of gradient descent method by a factor of $ε^{-0.25}$ and matching the best iteration complexity of second-order Hessian-free methods for non-convex optimization.

math.OC↗

arXiv2017

Improved Dynamic Regret for Non-degenerate Functions

Recently, there has been a growing research interest in the analysis of dynamic regret, which measures the performance of an online learner against a sequence of local minimizers. By exploiting the strong convexity, previous studies have shown that the dynamic regret can be upper bounded by the path-length of the comparator sequence. In this paper, we illustrate that the dynamic regret can be further improved by allowing the learner to query the gradient of the function multiple times, and meanwhile the strong convexity can be weakened to other non-degenerate conditions. Specifically, we introduce the squared path-length, which could be much smaller than the path-length, as a new regularity of the comparator sequence. When multiple gradients are accessible to the learner, we first demonstrate that the dynamic regret of strongly convex functions can be upper bounded by the minimum of the path-length and the squared path-length. We then extend our theoretical guarantee to functions that are semi-strongly convex or self-concordant. To the best of our knowledge, this is the first time that semi-strong convexity and self-concordance are utilized to tighten the dynamic regret.

cs.LG↗

arXiv2017

Extremely Low Bit Neural Network: Squeeze the Last Bit Out with ADMM

Although deep learning models are highly effective for various learning tasks, their high computational costs prohibit the deployment to scenarios where either memory or computational resources are limited. In this paper, we focus on compressing and accelerating deep models with network weights represented by very small numbers of bits, referred to as extremely low bit neural network. We model this problem as a discretely constrained optimization problem. Borrowing the idea from Alternating Direction Method of Multipliers (ADMM), we decouple the continuous parameters from the discrete constraints of network, and cast the original hard problem into several subproblems. We propose to solve these subproblems using extragradient and iterative quantization algorithms that lead to considerably faster convergency compared to conventional optimization methods. Extensive experiments on image recognition and object detection verify that the proposed algorithm is more effective than state-of-the-art approaches when coming to extremely low bit neural network.

cs.CV↗

arXiv2017

Empirical Risk Minimization for Stochastic Convex Optimization: $O(1/n)$- and $O(1/n^2)$-type of Risk Bounds

Although there exist plentiful theories of empirical risk minimization (ERM) for supervised learning, current theoretical understandings of ERM for a related problem---stochastic convex optimization (SCO), are limited. In this work, we strengthen the realm of ERM for SCO by exploiting smoothness and strong convexity conditions to improve the risk bounds. First, we establish an $\widetilde{O}(d/n + \sqrt{F_*/n})$ risk bound when the random function is nonnegative, convex and smooth, and the expected function is Lipschitz continuous, where $d$ is the dimensionality of the problem, $n$ is the number of samples, and $F_*$ is the minimal risk. Thus, when $F_*$ is small we obtain an $\widetilde{O}(d/n)$ risk bound, which is analogous to the $\widetilde{O}(1/n)$ optimistic rate of ERM for supervised learning. Second, if the objective function is also $λ$-strongly convex, we prove an $\widetilde{O}(d/n + κF_*/n )$ risk bound where $κ$ is the condition number, and improve it to $O(1/[λn^2] + κF_*/n)$ when $n=\widetildeΩ(κd)$. As a result, we obtain an $O(κ/n^2)$ risk bound under the condition that $n$ is large and $F_*$ is small, which to the best of our knowledge, is the first $O(1/n^2)$-type of risk bound of ERM. Third, we stress that the above results are established in a unified framework, which allows us to derive new risk bounds under weaker conditions, e.g., without convexity of the random function and Lipschitz continuity of the expected function. Finally, we demonstrate that to achieve an $O(1/[λn^2] + κF_*/n)$ risk bound for supervised learning, the $\widetildeΩ(κd)$ requirement on $n$ can be replaced with $Ω(κ^2)$, which is dimensionality-independent.

cs.LG↗

arXiv2016

Sparse Learning for Large-scale and High-dimensional Data: A Randomized Convex-concave Optimization Approach

In this paper, we develop a randomized algorithm and theory for learning a sparse model from large-scale and high-dimensional data, which is usually formulated as an empirical risk minimization problem with a sparsity-inducing regularizer. Under the assumption that there exists a (approximately) sparse solution with high classification accuracy, we argue that the dual solution is also sparse or approximately sparse. The fact that both primal and dual solutions are sparse motivates us to develop a randomized approach for a general convex-concave optimization problem. Specifically, the proposed approach combines the strength of random projection with that of sparse learning: it utilizes random projection to reduce the dimensionality, and introduces $\ell_1$-norm regularization to alleviate the approximation error caused by random projection. Theoretical analysis shows that under favored conditions, the randomized algorithm can accurately recover the optimal solutions to the convex-concave optimization problem (i.e., recover both the primal and dual solutions).

cs.LG↗

arXiv2016

A Simple Homotopy Proximal Mapping for Compressive Sensing

In this paper, we present a novel yet simple homotopy proximal mapping algorithm for compressive sensing. The algorithm adopts a simple proximal mapping of the $\ell_1$ norm at each iteration and gradually reduces the regularization parameter for the $\ell_1$ norm. We prove a global linear convergence of the proposed homotopy proximal mapping (HPM) algorithm for solving compressive sensing under three different settings (i) sparse signal recovery under noiseless measurements, (ii) sparse signal recovery under noisy measurements, and (iii) nearly-sparse signal recovery under sub-gaussian noisy measurements. In particular, we show that when the measurement matrix satisfies Restricted Isometric Properties (RIP), our theoretical results in settings (i) and (ii) almost recover the best condition on the RIP constants for compressive sensing. In addition, in setting (iii), our results for sparse signal recovery are better than the previous results, and furthermore our analysis explicitly exhibits that more observations lead to not only more accurate recovery but also faster convergence. Compared with previous studies on linear convergence for sparse signal recovery, our algorithm is simple and efficient, and our results are better and provide more insights. Finally our empirical studies provide further support for the proposed homotopy proximal mapping algorithm and verify the theoretical results.

cs.IT↗

arXiv2016

Tracking Slowly Moving Clairvoyant: Optimal Dynamic Regret of Online Learning with True and Noisy Gradient

This work focuses on dynamic regret of online convex optimization that compares the performance of online learning to a clairvoyant who knows the sequence of loss functions in advance and hence selects the minimizer of the loss function at each step. By assuming that the clairvoyant moves slowly (i.e., the minimizers change slowly), we present several improved variation-based upper bounds of the dynamic regret under the true and noisy gradient feedback, which are {\it optimal} in light of the presented lower bounds. The key to our analysis is to explore a regularity metric that measures the temporal changes in the clairvoyant's minimizers, to which we refer as {\it path variation}. Firstly, we present a general lower bound in terms of the path variation, and then show that under full information or gradient feedback we are able to achieve an optimal dynamic regret. Secondly, we present a lower bound with noisy gradient feedback and then show that we can achieve optimal dynamic regrets under a stochastic gradient feedback and two-point bandit feedback. Moreover, for a sequence of smooth loss functions that admit a small variation in the gradients, our dynamic regret under the two-point bandit feedback matches what is achieved with full information.

cs.LG↗

arXiv2015

Similarity Learning via Adaptive Regression and Its Application to Image Retrieval

We study the problem of similarity learning and its application to image retrieval with large-scale data. The similarity between pairs of images can be measured by the distances between their high dimensional representations, and the problem of learning the appropriate similarity is often addressed by distance metric learning. However, distance metric learning requires the learned metric to be a PSD matrix, which is computational expensive and not necessary for retrieval ranking problem. On the other hand, the bilinear model is shown to be more flexible for large-scale image retrieval task, hence, we adopt it to learn a matrix for estimating pairwise similarities under the regression framework. By adaptively updating the target matrix in regression, we can mimic the hinge loss, which is more appropriate for similarity learning problem. Although the regression problem can have the closed-form solution, the computational cost can be very expensive. The computational challenges come from two aspects: the number of images can be very large and image features have high dimensionality. We address the first challenge by compressing the data by a randomized algorithm with the theoretical guarantee. For the high dimensional issue, we address it by taking low rank assumption and applying alternating method to obtain the partial matrix, which has a global optimal solution. Empirical studies on real world image datasets (i.e., Caltech and ImageNet) demonstrate the effectiveness and efficiency of the proposed method.

cs.LG↗

arXiv2015

Stochastic Proximal Gradient Descent for Nuclear Norm Regularization

In this paper, we utilize stochastic optimization to reduce the space complexity of convex composite optimization with a nuclear norm regularizer, where the variable is a matrix of size $m \times n$. By constructing a low-rank estimate of the gradient, we propose an iterative algorithm based on stochastic proximal gradient descent (SPGD), and take the last iterate of SPGD as the final solution. The main advantage of the proposed algorithm is that its space complexity is $O(m+n)$, in contrast, most of previous algorithms have a $O(mn)$ space complexity. Theoretical analysis shows that it achieves $O(\log T/\sqrt{T})$ and $O(\log T/T)$ convergence rates for general convex functions and strongly convex functions, respectively.

cs.LG↗

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.

math.NA↗

arXiv2015

Online Stochastic Linear Optimization under One-bit Feedback

In this paper, we study a special bandit setting of online stochastic linear optimization, where only one-bit of information is revealed to the learner at each round. This problem has found many applications including online advertisement and online recommendation. We assume the binary feedback is a random variable generated from the logit model, and aim to minimize the regret defined by the unknown linear function. Although the existing method for generalized linear bandit can be applied to our problem, the high computational cost makes it impractical for real-world problems. To address this challenge, we develop an efficient online learning algorithm by exploiting particular structures of the observation model. Specifically, we adopt online Newton step to estimate the unknown parameter and derive a tight confidence region based on the exponential concavity of the logistic loss. Our analysis shows that the proposed algorithm achieves a regret bound of $O(d\sqrt{T})$, which matches the optimal result of stochastic linear bandits.

cs.LG↗

arXiv2015

Towards Making High Dimensional Distance Metric Learning Practical

In this work, we study distance metric learning (DML) for high dimensional data. A typical approach for DML with high dimensional data is to perform the dimensionality reduction first before learning the distance metric. The main shortcoming of this approach is that it may result in a suboptimal solution due to the subspace removed by the dimensionality reduction method. In this work, we present a dual random projection frame for DML with high dimensional data that explicitly addresses the limitation of dimensionality reduction for DML. The key idea is to first project all the data points into a low dimensional space by random projection, and compute the dual variables using the projected vectors. It then reconstructs the distance metric in the original space using the estimated dual variables. The proposed method, on one hand, enjoys the light computation of random projection, and on the other hand, alleviates the limitation of most dimensionality reduction methods. We verify both empirically and theoretically the effectiveness of the proposed algorithm for high dimensional DML.

cs.LG↗

arXiv2015

Theory of Dual-sparse Regularized Randomized Reduction

In this paper, we study randomized reduction methods, which reduce high-dimensional features into low-dimensional space by randomized methods (e.g., random projection, random hashing), for large-scale high-dimensional classification. Previous theoretical results on randomized reduction methods hinge on strong assumptions about the data, e.g., low rank of the data matrix or a large separable margin of classification, which hinder their applications in broad domains. To address these limitations, we propose dual-sparse regularized randomized reduction methods that introduce a sparse regularizer into the reduced dual problem. Under a mild condition that the original dual solution is a (nearly) sparse vector, we show that the resulting dual solution is close to the original dual solution and concentrates on its support set. In numerical experiments, we present an empirical study to support the analysis and we also present a novel application of the dual-sparse regularized randomized reduction methods to reducing the communication cost of distributed learning from large-scale high-dimensional data.

cs.LG↗

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.

math.ST↗

arXiv2015

Fine-Grained Visual Categorization via Multi-stage Metric Learning

Fine-grained visual categorization (FGVC) is to categorize objects into subordinate classes instead of basic classes. One major challenge in FGVC is the co-occurrence of two issues: 1) many subordinate classes are highly correlated and are difficult to distinguish, and 2) there exists the large intra-class variation (e.g., due to object pose). This paper proposes to explicitly address the above two issues via distance metric learning (DML). DML addresses the first issue by learning an embedding so that data points from the same class will be pulled together while those from different classes should be pushed apart from each other; and it addresses the second issue by allowing the flexibility that only a portion of the neighbors (not all data points) from the same class need to be pulled together. However, feature representation of an image is often high dimensional, and DML is known to have difficulty in dealing with high dimensional feature vectors since it would require $\mathcal{O}(d^2)$ for storage and $\mathcal{O}(d^3)$ for optimization. To this end, we proposed a multi-stage metric learning framework that divides the large-scale high dimensional learning problem to a series of simple subproblems, achieving $\mathcal{O}(d)$ computational complexity. The empirical study with FVGC benchmark datasets verifies that our method is both effective and efficient compared to the state-of-the-art FGVC approaches.

cs.CV↗

arXiv2015

An Explicit Sampling Dependent Spectral Error Bound for Column Subset Selection

In this paper, we consider the problem of column subset selection. We present a novel analysis of the spectral norm reconstruction for a simple randomized algorithm and establish a new bound that depends explicitly on the sampling probabilities. The sampling dependent error bound (i) allows us to better understand the tradeoff in the reconstruction error due to sampling probabilities, (ii) exhibits more insights than existing error bounds that exploit specific probability distributions, and (iii) implies better sampling distributions. In particular, we show that a sampling distribution with probabilities proportional to the square root of the statistical leverage scores is always better than uniform sampling and is better than leverage-based sampling when the statistical leverage scores are very nonuniform. And by solving a constrained optimization problem related to the error bound with an efficient bisection search we are able to achieve better performance than using either the leverage-based distribution or that proportional to the square root of the statistical leverage scores. Numerical simulations demonstrate the benefits of the new sampling distributions for low-rank matrix approximation and least square approximation compared to state-of-the art algorithms.

math.NA↗

arXiv2014

CUR Algorithm for Partially Observed Matrices

CUR matrix decomposition computes the low rank approximation of a given matrix by using the actual rows and columns of the matrix. It has been a very useful tool for handling large matrices. One limitation with the existing algorithms for CUR matrix decomposition is that they need an access to the {\it full} matrix, a requirement that can be difficult to fulfill in many real world applications. In this work, we alleviate this limitation by developing a CUR decomposition algorithm for partially observed matrices. In particular, the proposed algorithm computes the low rank approximation of the target matrix based on (i) the randomly sampled rows and columns, and (ii) a subset of observed entries that are randomly sampled from the matrix. Our analysis shows the relative error bound, measured by spectral norm, for the proposed algorithm when the target matrix is of full rank. We also show that only $O(n r\ln r)$ observed entries are needed by the proposed algorithm to perfectly recover a rank $r$ matrix of size $n\times n$, which improves the sample complexity of the existing algorithms for matrix completion. Empirical studies on both synthetic and real-world datasets verify our theoretical claims and demonstrate the effectiveness of the proposed algorithm.

cs.LG↗

arXiv2014

Top Rank Optimization in Linear Time

Bipartite ranking aims to learn a real-valued ranking function that orders positive instances before negative instances. Recent efforts of bipartite ranking are focused on optimizing ranking accuracy at the top of the ranked list. Most existing approaches are either to optimize task specific metrics or to extend the ranking loss by emphasizing more on the error associated with the top ranked instances, leading to a high computational cost that is super-linear in the number of training instances. We propose a highly efficient approach, titled TopPush, for optimizing accuracy at the top that has computational complexity linear in the number of training instances. We present a novel analysis that bounds the generalization error for the top ranked instances for the proposed approach. Empirical study shows that the proposed approach is highly competitive to the state-of-the-art approaches and is 10-100 times faster.

cs.LG↗