In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter. In this paper, we propose a continuation algorithm that is applicable to a large class of nonsmooth regularized risk minimization problems, can be flexibly used with a number of existing solvers for the underlying smoothed subproblem, and with convergence results on the whole algorithm rather than just one of its subproblems. In particular, when accelerated solvers are used, the proposed algorithm achieves the fastest known rates of @math on strongly convex problems, and @math on general convex problems. Experiments on nonsmooth classification and regression tasks demonstrate that the proposed algorithm outperforms the state-of-the-art.
For example, consider the hinge loss @math , where @math is the linear model parameter, and @math is the @math th training sample with @math . Using @math , @math can be smoothed to @cite_12 .. Similarly, the @math loss @math can be smoothed to .. Other examples in machine learning include popular regularizers such as the @math , total variation @cite_1 , overlapping group lasso, and graph-guided fused lasso @cite_26 .
@cite_26 @cite_1 @cite_12
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We study the problem of estimating high-dimensional regression models regularized by a structured sparsity-inducing penalty that encodes prior structural information on either the input or output variables. We consider two widely adopted types of penalties of this kind as motivating examples: (1) the general overlapping-group-lasso penalty, generalized from the group-lasso penalty; and (2) the graph-guided-fused-lasso penalty, generalized from the fused-lasso penalty. For both types of penalties, due to their nonseparability and nonsmoothness, developing an efficient optimization method remains a challenging problem. In this paper we propose a general optimization approach, the smoothing proximal gradient (SPG) method, which can solve structured sparse regression problems with any smooth convex loss under a wide spectrum of structured sparsity-inducing penalties. Our approach combines a smoothing technique with an effective proximal gradient method. It achieves a convergence rate significantly faster than the standard first-order methods, subgradient methods, and is much more scalable than the most widely used interior-point methods. The efficiency and scalability of our method are demonstrated on both simulation experiments and real genetic data sets. Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. This paper applies a smoothing technique and an accelerated first-order algorithm, both from Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127-152], and demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as (1) it is computationally efficient, (2) it is accurate and returns solutions with several correct digits, (3) it is flexible and amenable to many kinds of reconstruction problems, and (4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization and convex programs seeking to minimize the @math norm of @math under constraints, in which @math is not diagonal. The code is available online as a free package in the MATLAB language. In this work we consider the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We propose a novel algorithm called Accelerated Nonsmooth Stochastic Gradient Descent (ANSGD), which exploits the structure of common nonsmooth loss functions to achieve optimal convergence rates for a class of problems including SVMs. It is the first stochastic algorithm that can achieve the optimal O(1 t) rate for minimizing nonsmooth loss functions (with strong convexity). The fast rates are confirmed by empirical comparisons, in which ANSGD significantly outperforms previous subgradient descent algorithms including SGD.