We propose an algorithm-independent framework to equip existing optimization methods with primal-dual certificates. Such certificates and corresponding rate of convergence guarantees are important for practitioners to diagnose progress, in particular in machine learning applications. We obtain new primal-dual convergence rates, e.g., for the Lasso as well as many L1, Elastic Net, group Lasso and TV-regularized problems. The theory applies to any norm-regularized generalized linear model. Our approach provides efficiently computable duality gaps which are globally defined, without modifying the original problems in the region of interest.
@cite_22: We investigate the relation of two fundamental tools in machine learning and signal processing, that is the support vector machine (SVM) for classification, and the Lasso technique used in regression. We show that the resulting optimization problems are equivalent, in the following sense. Given any instance of an L2-loss soft-margin (or hard-margin) SVM, we construct a Lasso instance having the same optimal solutions, and vice versa. As a consequence, many existing optimization algorithms for both SVMs and Lasso can also be applied to the respective other problem instances. Also, the equivalence allows for many known theoretical insights for SVM and Lasso to be translated between the two settings. One such implication gives a simple kernelized version of the Lasso, analogous to the kernels used in the SVM setting. Another consequence is that the sparsity of a Lasso solution is equal to the number of support vectors for the corresponding SVM instance, and that one can use screening rules to prune the set of support vectors. Furthermore, we can relate sublinear time algorithms for the two problems, and give a new such algorithm variant for the Lasso. We also study the regularization paths for both methods.
@cite_23: We describe and analyze two stochastic methods for l1 regularized loss minimization problems, such as the Lasso. The first method updates the weight of a single feature at each iteration while the second method updates the entire weight vector but only uses a single training example at each iteration. In both methods, the choice of feature or example is uniformly at random. Our theoretical runtime analysis suggests that the stochastic methods should outperform state-of-the-art deterministic approaches, including their deterministic counterparts, when the size of the problem is large. We demonstrate the advantage of stochastic methods by experimenting with synthetic and natural data sets.
@cite_15: We develop fast algorithms for estimation of generalized linear models with convex penalties. The models include linear regression, two-class logistic regression, and multi- nomial regression problems while the penalties include l1 (the lasso), l2 (ridge regression) and mixtures of the two (the elastic net). The algorithms use cyclical coordinate descent, computed along a regularization path. The methods can handle large problems and can also deal efficiently with sparse features. In comparative timings we find that the new algorithms are considerably faster than competing methods.
@cite_16: Large-scale linear classification is widely used in many areas. The L1-regularized form can be applied for feature selection; however, its non-differentiability causes more difficulties in training. Although various optimization methods have been proposed in recent years, these have not yet been compared suitably. In this paper, we first broadly review existing methods. Then, we discuss state-of-the-art software packages in detail and propose two efficient implementations. Extensive comparisons indicate that carefully implemented coordinate descent methods are very suitable for training large document data.
@cite_12: Recently, (2010) conducted a comprehensive comparison on software for L1-regularized classification. They concluded that a carefully designed coordinate descent implementation CDN is the fastest among state-of-the-art solvers. In this paper, we point out that CDN is less competitive on loss functions that are expensive to compute. In particular, CDN for logistic regression is much slower than CDN for SVM because the logistic loss involves expensive exp log operations. In optimization, Newton methods are known to have fewer iterations although each iteration costs more. Because solving the Newton sub-problem is independent of the loss calculation, this type of methods may surpass CDN under some circumstances. In L1-regularized classification, GLMNET by is already a Newton-type method, but experiments in (2010) indicated that the existing GLMNET implementation may face difficulties for some largescale problems. In this paper, we propose an improved GLMNET to address some theoretical and implementation issues. In particular, as a Newton-type method, GLMNET achieves fast local convergence, but may fail to quickly obtain a useful solution. By a careful design to adjust the effort for each iteration, our method is efficient for both loosely or strictly solving the optimization problem. Experiments demonstrate that our improved GLMNET ismore efficient than CDN for L1-regularized logistic regression.
In the Lasso literature, a similar trend in terms of solvers has been observed recently, but with the roles of primal and dual problems reversed. For those problems, coordinate descent algorithms on the primal formulation have become the state-of-the-art, as in @cite_15 and extensions @cite_23 @cite_12 @cite_16 . Despite the prototypes of both problem types---SVM for the @math -regularized case and Lasso for the @math -regularized case---being closely related @cite_22 , we are not aware of existing primal-dual guarantees for coordinate descent on unmodified @math -regularized problems.