Generating adversarial examples (AEs) can be formulated as an optimization problem. Among various optimization-based attacks, the gradient-based PGD and the momentum-based MI-FGSM have garnered considerable interest. However, all these attacks use the sign function to scale their perturbations, which raises several theoretical concerns from the point of view of optimization. In this paper, we first reveal that PGD is actually a specific reformulation of the projected gradient method using only the current gradient to determine its step-size. Further, we show that when we utilize a conventional adaptive matrix with the accumulated gradients to scale the perturbation, PGD becomes AdaGrad. Motivated by this analysis, we present a novel momentum-based attack AdaMI, in which the perturbation is optimized with an interesting momentum-based adaptive matrix. AdaMI is proved to attain optimal convergence for convex problems, indicating that it addresses the non-convergence issue of MI-FGSM, thereby ensuring stability of the optimization process. The experiments demonstrate that the proposed momentum-based adaptive matrix can serve as a general and effective technique to boost adversarial transferability over the state-of-the-art methods across different networks while maintaining better stability and imperceptibility.

Doubly stochastic matrix plays an essential role in several areas such as statistics and machine learning. In this paper we consider the optimal approximation of a square matrix in the set of doubly stochastic matrices. A structured BFGS method is proposed to solve the dual of the primal problem. The resulting algorithm builds curvature information into the diagonal components of the true Hessian, so that it takes only additional linear cost to obtain the descent direction based on the gradient information without having to explicitly store the inverse Hessian approximation. The cost is substantially fewer than quadratic complexity of the classical BFGS algorithm. Meanwhile, a Newton-based line search method is presented for finding a suitable step size, which in practice uses the existing knowledge and takes only one iteration. The global convergence of our algorithm is established. We verify the advantages of our approach on both synthetic data and real data sets. The experimental results demonstrate that our algorithm outperforms the state-of-the-art solvers and enjoys outstanding scalability.

Momentum technique has recently emerged as an effective strategy in accelerating convergence of gradient descent (GD) methods and exhibits improved performance in deep learning as well as regularized learning. Typical momentum examples include Nesterov's accelerated gradient (NAG) and heavy-ball (HB) methods. However, so far, almost all the acceleration analyses are only limited to NAG, and a few investigations about the acceleration of HB are reported. In this article, we address the convergence about the last iterate of HB in nonsmooth optimizations with constraints, which we name individual convergence. This question is significant in machine learning, where the constraints are required to impose on the learning structure and the individual output is needed to effectively guarantee this structure while keeping an optimal rate of convergence. Specifically, we prove that HB achieves an individual convergence rate of O(1/√t) , where t is the number of iterations. This indicates that both of the two momentum methods can accelerate the individual convergence of basic GD to be optimal. Even for the convergence of averaged iterates, our result avoids the disadvantages of the previous work in restricting the optimization problem to be unconstrained as well as limiting the performed number of iterations to be predefined. The novelty of convergence analysis presented in this article provides a clear understanding of how the HB momentum can accelerate the individual convergence and reveals more insights about the similarities and differences in getting the averaging and individual convergence rates. The derived optimal individual convergence is extended to regularized and stochastic settings, in which an individual solution can be produced by the projection-based operation. In contrast to the averaged output, the sparsity can be reduced remarkably without sacrificing the theoretical optimal rates. Several real experiments demonstrate the performance of HB momentum strategy.

Averaging scheme has attracted extensive attention in deep learning as well as traditional machine learning. It achieves theoretically optimal convergence and also improves the empirical model performance. However, there is still a lack of sufficient convergence analysis for strongly convex optimization. Typically, the convergence about the last iterate of gradient descent methods, which is referred to as individual convergence, fails to attain its optimality due to the existence of logarithmic factor. In order to remove this factor, we first develop gradient descent averaging (GDA), which is a general projection-based dual averaging algorithm in the strongly convex setting. We further present primal-dual averaging for strongly convex cases (SC-PDA), where primal and dual averaging schemes are simultaneously utilized. We prove that GDA yields the optimal convergence rate in terms of output averaging, while SC-PDA derives the optimal individual convergence. Several experiments on SVMs and deep learning models validate the correctness of theoretical analysis and effectiveness of algorithms.

Many well-known first-order gradient methods have been extended to cope with large-scale composite problems, which often arise as a regularized empirical risk minimization in machine learning. However, their optimal convergence is attained only in terms of the weighted average of past iterative solutions. How to make the individual convergence of stochastic gradient descent (SGD) optimal, especially for strongly convex problems has now become a challenging problem in the machine learning community. On the other hand, Nesterov's recent weighted averaging strategy succeeds in achieving the optimal individual convergence of dual averaging (DA) but it fails in the basic mirror descent (MD). In this paper, a new primal averaging (PA) gradient operation step is presented, in which the gradient evaluation is imposed on the weighted average of all past iterative solutions. We prove that simply modifying the gradient operation step in MD by PA strategy suffices to recover the optimal individual rate for general convex problems. Along this line, the optimal individual rate of convergence for strongly convex problems can also be achieved by imposing the strong convexity on the gradient operation step. Furthermore, we extend PA-MD to solve regularized nonsmooth learning problems in the stochastic setting, which reveals that PA strategy is a simple yet effective extra step toward the optimal individual convergence of SGD. Several real experiments on sparse learning and SVM problems verify the correctness of our theoretical analysis.

The extrapolation strategy raised by Nesterov, which can accelerate the convergence rate of gradient descent methods by orders of magnitude when dealing with smooth convex objective, has led to tremendous success in training machine learning tasks. In this article, the convergence of individual iterates of projected subgradient (PSG) methods for nonsmooth convex optimization problems is theoretically studied based on Nesterov's extrapolation, which we name individual convergence. We prove that Nesterov's extrapolation has the strength to make the individual convergence of PSG optimal for nonsmooth problems. In light of this consideration, a direct modification of the subgradient evaluation suffices to achieve optimal individual convergence for strongly convex problems, which can be regarded as making an interesting step toward the open question about stochastic gradient descent (SGD) posed by Shamir. Furthermore, we give an extension of the derived algorithms to solve regularized learning tasks with nonsmooth losses in stochastic settings. Compared with other state-of-the-art nonsmooth methods, the derived algorithms can serve as an alternative to the basic SGD especially in coping with machine learning problems, where an individual output is needed to guarantee the regularization structure while keeping an optimal rate of convergence. Typically, our method is applicable as an efficient tool for solving large-scale l1-regularized hinge-loss learning problems. Several comparison experiments demonstrate that our individual output not only achieves an optimal convergence rate but also guarantees better sparsity than the averaged solution.

Top- performance has recently received increasing attention in large data categories. Advances, like a top- multiclass support vector machine (SVM), have consistently improved the top- accuracy. However, the key ingredient in the state-of-the-art optimization scheme based upon stochastic dual coordinate ascent relies on the sorting method, which yields complexity. In this paper, we leverage the semismoothness of the problem and propose an optimized top- multiclass SVM algorithm, which employs semismooth Newton algorithm for the key building block to improve the training speed. Our method enjoys a local superlinear convergence rate in theory. In practice, experimental results confirm the validity. Our algorithm is four times faster than the existing method in large synthetic problems; Moreover, on real-world data sets it also shows significant improvement in training time.

The truncated regular -loss support vector machine can eliminate the excessive number of support vectors (SVs); thus, it has significant advantages in robustness and scalability. However, in this paper, we discover that the associated state-of-the-art solvers, such as difference convex algorithm and concave-convex procedure, not only have limited sparsity promoting property for general truncated losses especially the -loss but also have poor scalability for large-scale problems. To circumvent these drawbacks, we present a general multistage scheme with explicit interpretation regarding SVs as well as outliers. In particular, we solve the general nonconvex truncated loss minimization through a sequence of associated convex subproblems, in which the outliers are removed in advance. The proposed algorithm can be regarded as a structural optimization attempt carefully considering sparsity imposed by the nonconvex truncated losses. We show that this general multistage algorithm offers sufficient sparsity especially for the truncated -loss. To further improve the scalability, we propose a linear multistep algorithm by employing a single iteration of coordinate descent to monotonically decrease the objective function at each stage and a kernel algorithm by using the Karush-Kuhn-Tucker conditions to cheaply find most part of the outliers for the next stage. Comparison experiments demonstrate that our methods have superiority in sparsity as well as efficiency in scalability.