arXiv subjects

Yunwen Lei

Publications and source records attributed to Yunwen Lei.

47 records · Page 3Linked to original sources

arXiv2019

Convergence of Online Mirror Descent

In this paper we consider online mirror descent (OMD) algorithms, a class of scalable online learning algorithms exploiting data geometric structures through mirror maps. Necessary and sufficient conditions are presented in terms of the step size sequence $\{η_t\}_{t}$ for the convergence of an OMD algorithm with respect to the expected Bregman distance induced by the mirror map. The condition is $\lim_{t\to\infty}η_t=0, \sum_{t=1}^{\infty}η_t=\infty$ in the case of positive variances. It is reduced to $\sum_{t=1}^{\infty}η_t=\infty$ in the case of zero variances for which the linear convergence may be achieved by taking a constant step size sequence. A sufficient condition on the almost sure convergence is also given. We establish tight error bounds under mild conditions on the mirror map, the loss function, and the regularizer. Our results are achieved by some novel analysis on the one-step progress of the OMD algorithm using smoothness and strong convexity of the mirror map and the loss function.

cs.LG↗

arXiv2019

Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions

Stochastic gradient descent (SGD) is a popular and efficient method with wide applications in training deep neural nets and other nonconvex models. While the behavior of SGD is well understood in the convex learning setting, the existing theoretical results for SGD applied to nonconvex objective functions are far from mature. For example, existing results require to impose a nontrivial assumption on the uniform boundedness of gradients for all iterates encountered in the learning process, which is hard to verify in practical implementations. In this paper, we establish a rigorous theoretical foundation for SGD in nonconvex learning by showing that this boundedness assumption can be removed without affecting convergence rates. In particular, we establish sufficient conditions for almost sure convergence as well as optimal convergence rates for SGD applied to both general nonconvex objective functions and gradient-dominated objective functions. A linear convergence is further derived in the case with zero variances.

cs.LG↗

arXiv2019

On Performance Estimation in Automatic Algorithm Configuration

Over the last decade, research on automated parameter tuning, often referred to as automatic algorithm configuration (AAC), has made significant progress. Although the usefulness of such tools has been widely recognized in real world applications, the theoretical foundations of AAC are still very weak. This paper addresses this gap by studying the performance estimation problem in AAC. More specifically, this paper first proves the universal best performance estimator in a practical setting, and then establishes theoretical bounds on the estimation error, i.e., the difference between the training performance and the true performance for a parameter configuration, considering finite and infinite configuration spaces respectively. These findings were verified in extensive experiments conducted on four algorithm configuration scenarios involving different problem domains. Moreover, insights for enhancing existing AAC methods are also identified.

cs.LG↗

arXiv2019

Stochastic Proximal AUC Maximization

In this paper we consider the problem of maximizing the Area under the ROC curve (AUC) which is a widely used performance metric in imbalanced classification and anomaly detection. Due to the pairwise nonlinearity of the objective function, classical SGD algorithms do not apply to the task of AUC maximization. We propose a novel stochastic proximal algorithm for AUC maximization which is scalable to large scale streaming data. Our algorithm can accommodate general penalty terms and is easy to implement with favorable $O(d)$ space and per-iteration time complexities. We establish a high-probability convergence rate $O(1/\sqrt{T})$ for the general convex setting, and improve it to a fast convergence rate $O(1/T)$ for the cases of strongly convex regularizers and no regularization term (without strong convexity). Our proof does not need the uniform boundedness assumption on the loss function or the iterates which is more fidelity to the practice. Finally, we perform extensive experiments over various benchmark data sets from real-world application domains which show the superior performance of our algorithm over the existing AUC maximization algorithms.

cs.LG↗

arXiv2019

A Generalization Error Bound for Multi-class Domain Generalization

Domain generalization is the problem of assigning labels to an unlabeled data set, given several similar data sets for which labels have been provided. Despite considerable interest in this problem over the last decade, there has been no theoretical analysis in the setting of multi-class classification. In this work, we study a kernel-based learning algorithm and establish a generalization error bound that scales logarithmically in the number of classes, matching state-of-the-art bounds for multi-class classification in the conventional learning setting. We also demonstrate empirically that the proposed algorithm achieves significant performance gains compared to a pooling strategy.

stat.ML↗

arXiv2017

Data-dependent Generalization Bounds for Multi-class Classification

In this paper, we study data-dependent generalization error bounds exhibiting a mild dependency on the number of classes, making them suitable for multi-class learning with a large number of label classes. The bounds generally hold for empirical multi-class risk minimization algorithms using an arbitrary norm as regularizer. Key to our analysis are new structural results for multi-class Gaussian complexities and empirical $\ell_\infty$-norm covering numbers, which exploit the Lipschitz continuity of the loss function with respect to the $\ell_2$- and $\ell_\infty$-norm, respectively. We establish data-dependent error bounds in terms of complexities of a linear function class defined on a finite set induced by training examples, for which we show tight lower and upper bounds. We apply the results to several prominent multi-class learning machines, exhibiting a tighter dependency on the number of classes than the state of the art. For instance, for the multi-class SVM by Crammer and Singer (2002), we obtain a data-dependent bound with a logarithmic dependency which significantly improves the previous square-root dependency. Experimental results are reported to verify the effectiveness of our theoretical findings.

cs.LG↗

arXiv2017

Convergence of Unregularized Online Learning Algorithms

In this paper we study the convergence of online gradient descent algorithms in reproducing kernel Hilbert spaces (RKHSs) without regularization. We establish a sufficient condition and a necessary condition for the convergence of excess generalization errors in expectation. A sufficient condition for the almost sure convergence is also given. With high probability, we provide explicit convergence rates of the excess generalization errors for both averaged iterates and the last iterate, which in turn also imply convergence rates with probability one. To our best knowledge, this is the first high-probability convergence rate for the last iterate of online gradient descent algorithms without strong convexity. Without any boundedness assumptions on iterates, our results are derived by a novel use of two measures of the algorithm's one-step progress, respectively by generalization errors and by distances in RKHSs, where the variances of the involved martingales are cancelled out by the descent property of the algorithm.

cs.LG↗

arXiv2017

Local Rademacher Complexity-based Learning Guarantees for Multi-Task Learning

We show a Talagrand-type concentration inequality for Multi-Task Learning (MTL), using which we establish sharp excess risk bounds for MTL in terms of distribution- and data-dependent versions of the Local Rademacher Complexity (LRC). We also give a new bound on the LRC for norm regularized as well as strongly convex hypothesis classes, which applies not only to MTL but also to the standard i.i.d. setting. Combining both results, one can now easily derive fast-rate bounds on the excess risk for many prominent MTL methods, including---as we demonstrate---Schatten-norm, group-norm, and graph-regularized MTL. The derived bounds reflect a relationship akeen to a conservation law of asymptotic convergence rates. This very relationship allows for trading off slower rates w.r.t. the number of tasks for faster rates with respect to the number of available samples per task, when compared to the rates obtained via a traditional, global Rademacher analysis.

cs.LG↗

arXiv2016

Localized Multiple Kernel Learning---A Convex Approach

We propose a localized approach to multiple kernel learning that can be formulated as a convex optimization problem over a given cluster structure. For which we obtain generalization error guarantees and derive an optimization algorithm based on the Fenchel dual representation. Experiments on real-world datasets from the application domains of computational biology and computer vision show that convex localized multiple kernel learning can achieve higher prediction accuracies than its global and non-convex local counterparts.

cs.LG↗

arXiv2015

Local Rademacher Complexity Bounds based on Covering Numbers

This paper provides a general result on controlling local Rademacher complexities, which captures in an elegant form to relate the complexities with constraint on the expected norm to the corresponding ones with constraint on the empirical norm. This result is convenient to apply in real applications and could yield refined local Rademacher complexity bounds for function classes satisfying general entropy conditions. We demonstrate the power of our complexity bounds by applying them to derive effective generalization error bounds.

cs.AI↗

arXiv2015

Multi-class SVMs: From Tighter Data-Dependent Generalization Bounds to Novel Algorithms

This paper studies the generalization performance of multi-class classification algorithms, for which we obtain, for the first time, a data-dependent generalization error bound with a logarithmic dependence on the class size, substantially improving the state-of-the-art linear dependence in the existing data-dependent generalization analysis. The theoretical analysis motivates us to introduce a new multi-class classification machine based on $\ell_p$-norm regularization, where the parameter $p$ controls the complexity of the corresponding bounds. We derive an efficient optimization algorithm based on Fenchel duality theory. Benchmarks on several real-world datasets show that the proposed algorithm can achieve significant accuracy gains over the state of the art.

cs.LG↗