Many machine learning tasks can be formulated as Regularized Empirical Risk Minimization (R-ERM), and solved by optimization algorithms such as gradient descent (GD), stochastic gradient descent (SGD), and stochastic variance reduction (SVRG). Conventional analysis on these optimization algorithms focuses on their convergence rates during the training process, however, people in the machine learning community may care more about the generalization performance of the learned model on unseen test data. In this paper, we investigate on this issue, by using stability as a tool. In particular, we decompose the generalization error for R-ERM, and derive its upper bound for both convex and nonconvex cases. In convex cases, we prove that the generalization error can be bounded by the convergence rate of the optimization algorithm and the stability of the R-ERM process, both in expectation (in the order of 𝒪(1/ n )+ 𝔼ρ( T )), where ρ( T ) is the convergence error and T is the number of iterations) and in high probability (in the order of 𝒪(log{1/δ / √ n + ρ( T ) with probability 1 – δ). For nonconvex cases, we can also obtain a similar expected generalization error bound. Our theorems indicate that 1) along with the training process, the generalization error will decrease for all the optimization algorithms under our investigation; 2) Comparatively speaking, SVRG has better generalization ability than GD and SGD. We have conducted experiments on both convex and nonconvex problems, and the experimental results verify our theoretical findings.

Regularized empirical risk minimization (R-ERM) is an important branch of machine learning, since it constrains the capacity of the hypothesis space and guarantees the generalization ability of the learning algorithm. Two classic proximal optimization algorithms, i.e., proximal stochastic gradient descent (ProxSGD) and proximal stochastic coordinate descent (ProxSCD) have been widely used to solve the R-ERM problem. Recently, variance reduction technique was proposed to improve ProxSGD and ProxSCD, and the corresponding ProxSVRG and ProxSVRCD have better convergence rate. These proximal algorithms with variance reduction technique have also achieved great success in applications at small and moderate scales. However, in order to solve large-scale R-ERM problems and make more practical impacts, the parallel versions of these algorithms are sorely needed. In this paper, we propose asynchronous ProxSVRG (Async-ProxSVRG) and asynchronous ProxSVRCD (Async-ProxSVRCD) algorithms, and prove that Async-ProxSVRG can achieve near linear speedup when the training data is sparse, while Async-ProxSVRCD can achieve near linear speedup regardless of the sparse condition, as long as the number of block partitions are appropriately set. We have conducted experiments on a regularized logistic regression task. The results verified our theoretical findings and demonstrated the practical efficiency of the asynchronous stochastic proximal algorithms with variance reduction.

Recently, many variance reduced stochastic alternating direction method of multipliers (ADMM) methods (e.g. SAG-ADMM, SDCA-ADMM and SVRG-ADMM) have made exciting progress such as linear convergence rates for strongly convex problems. However, the best known convergence rate for general convex problems is O(1/ T ) as opposed to O(1/ T 2 ) of accelerated batch algorithms, where T is the number of iterations. Thus, there still remains a gap in convergence rates between existing stochastic ADMM and batch algorithms. To bridge this gap, we introduce the momentum acceleration trick for batch optimization into the stochastic variance reduced gradient based ADMM (SVRG-ADMM), which leads to an accelerated (ASVRG-ADMM) method. Then we design two different momentum term update rules for strongly convex and general convex cases. We prove that ASVRG-ADMM converges linearly for strongly convex problems. Besides having a low-iteration complexity as existing stochastic ADMM methods, ASVRG-ADMM improves the convergence rate on general convex problems from O(1/ T ) to O(1/T 2 ). Our experimental results show the effectiveness of ASVRG-ADMM.

Kernel learning is a fundamental problem both in recent research and application of kernel methods. Existing kernel learning methods commonly use some measures of generalization errors to learn the optimal kernel in a convex (or conic) combination of prescribed basic kernels. However, the generalization bounds derived by these measures usually have slow convergence rates, and the basic kernels are finite and should be specified in advance. In this paper, we propose a new kernel learning method based on a novel measure of generalization error, called principal eigenvalue proportion (PEP), which can learn the optimal kernel with sharp generalization bounds over the convex hull of a possibly infinite set of basic kernels. We first derive sharp generalization bounds based on the PEP measure. Then we design two kernel learning algorithms for finite kernels and infinite kernels respectively, in which the derived sharp generalization bounds are exploited to guarantee faster convergence rates, moreover, basic kernels can be learned automatically for infinite kernel learning instead of being prescribed in advance. Theoretical analysis and empirical results demonstrate that the proposed kernel learning method outperforms the state-of-the-art kernel learning methods.

Extensive studies have demonstrated that the representations of convolutional neural networks (CNN), which are learned from a large-scale data set in the source domain, can be effectively transferred to a new target domain. However, compared to the source domain, the target domain often has limited data in practice. In this case, overfitting may significantly depress transferability, due to the model redundancy of the intensive CNN structures. To deal with this difficulty, we propose a novel sparse deep transfer learning approach for CNN. There are three main contributions in this work. First, we introduce a Sparse-SourceNet to reduce the redundancy in the source domain. Second, we introduce a Hybrid-TransferNet to improve the generalization ability and the prediction accuracy of transfer learning, by taking advantage of both model sparsity and implicit knowledge. Third, we introduce a Sparse-TargetNet, where we prune our Hybrid-TransferNet to obtain a highly-compact, source-knowledge-integrated CNN in the target domain. To examine the effectiveness of our methods, we perform our sparse deep transfer learning approach on a number of benchmark transfer learning tasks. The results show that, compared to the standard fine-tuning approach, our proposed approach achieves a significant pruning rate on CNN while improves the accuracy of transfer learning.

Recent years have witnessed extensive attention in binary code learning, a.k.a. hashing, for nearest neighbor search problems. It has been seen that high-dimensional data points can quantize into binary codes to give an efficient similarity approximation via Hamming distance. Among the existing schemes, ranking-based hashing is recent promising that targets at preserving ordinal relations of ranking in the Hamming space to minimize retrieval loss. However, the size of the ranking tuples that show the ordinal relations, is quadratic or cubic to the size of training samples. It is so very expensive to embed such ranking tuples in binary code learning, especially given a large-scale training data set. Besides, it remains difficult to build ranking tuples efficiently for most ranking-preserving hashing, which are deployed over an ordinal graph-based setting. To handle these problems, we propose a novel ranking-preserving hashing method, dubbed Ordinal Constraint Hashing (OCH), which efficiently learns the optimal hashing functions with a graph-based approximation to embed the ordinal relations. The core idea is to reduce the size of ordinal graph with ordinal constraint projection, which preserves the ordinal relations through a small data set (such as clusters or random samples). In particular, to learn such hash functions effectively, we further relax the discrete constraints and design a specific stochastic gradient decent algorithm for optimization. Experimental results on three large-scale visual search benchmark datasets, i.e. LabelMe, Tiny100K and GIST1M, show that the proposed OCH method can achieve superior performance over the state-of-the-arts approaches.

Semi-supervised learning (SSL) concerns how to improve performance via the usage of unlabeled data. Recent studies indicate that the usage of unlabeled data might even deteriorate performance. Although some proposals have been developed to alleviate such a fundamental challenge for semi-supervised classification, the efforts on semi-supervised regression (SSR) remain to be limited. In this work we consider the learning of a safe prediction from multiple semi-supervised regressors, which is not worse than a direct supervised learner with only labeled data. We cast it as a geometric projection issue with an efficient algorithm. Furthermore, we show that the proposal is provably safe and has already achieved the maximal performance gain, if the ground-truth label assignment is realized by a convex linear combination of base regressors. This provides insight to help understand safe SSR. Experimental results on a broad range of datasets validate the effectiveness of our proposal.

Faced with the requirements of huge amounts of data processing nowadays, hashing techniques have attracted much attention due to their efficient storage and searching ability. Among these techniques, the ones based on spectral graph show remarkable performance as they could embed the data on a low-dimensional manifold and maintain the neighborhood structure via a non-linear spectral eigenmap. However, the spectral solution in real value of such methods may deviate from the discrete solution. The common practice is just performing a simple rounding operation to obtain the final binary codes, which could break constraints and even result in worse condition. In this paper, we propose to impose a so-called spectral rotation technique to the spectral hashing objective, which could transform the candidate solution into a new one that better approximates the discrete one. Moreover, the binary codes are obtained from the modified solution via minimizing the Euclidean distance, which could result in more semantical correlation within the manifold, where the constraints for codes are always held. We provide an efficient alternative algorithm to solve the above problems. And a manifold learning perceptive for motivating the proposed method is also shown. Extensive experiments are conducted on three large-scale benchmark datasets and the results show our method outperforms state-of-the-art hashing methods, especially the spectral graph ones.

We propose a method that involves a probabilistic model for learning future classifiers for tasks in which decision boundaries nonlinearly change over time. In certain applications, such as spam-mail classification, the decision boundary dynamically changes over time. Accordingly, the performance of the classifiers will deteriorate quickly unless the classifiers are updated using additional data. However, collecting such data can be expensive or impossible. The proposed model alleviates this deterioration in performance without additional data by modeling the non-linear dynamics of the decision boundary using Gaussian processes. The method also involves our developed learning algorithm for our model based on empirical variational Bayesian inference by which uncertainty of dynamics can be incorporated for future classification. The effectiveness of the proposed method was demonstrated through experiments using synthetic and real-world data sets.

Gaussian state space models have been used for decades as generative models of sequential data. They admit an intuitive probabilistic interpretation, have a simple functional form, and enjoy widespread adoption. We introduce a unified algorithm to efficiently learn a broad class of linear and non-linear state space models, including variants where the emission and transition distributions are modeled by deep neural networks. Our learning algorithm simultaneously learns a compiled inference network and the generative model, leveraging a structured variational approximation parameterized by recurrent neural networks to mimic the posterior distribution. We apply the learning algorithm to both synthetic and real-world datasets, demonstrating its scalability and versatility. We find that using the structured approximation to the posterior results in models with significantly higher held-out likelihood.