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Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an ε -stationary point of the objective function in O ε - 2 (resp. O ε - 4 ) iterations under nonconvex-strongly concave (resp. nonconvex–concave) setting. Moreover, its gradient complexity to obtain an ε -stationary point of the objective function is bounded by O ε - 2 (resp., O ε - 4 ) under the strongly convex–nonconcave (resp., convex–nonconcave) setting. To the best of our knowledge, this is the first time that a simple and unified single-loop algorithm is developed for solving both nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. Moreover, the complexity results for solving the latter (strongly) convex–nonconcave minimax problems have never been obtained before in the literature. Numerical results show the efficiency of the proposed AGP algorithm. Furthermore, we extend the AGP algorithm by presenting a block alternating proximal gradient (BAPG) algorithm for solving more general multi-block nonsmooth nonconvex-(strongly) concave and (strongly) convex–nonconcave minimax problems. We can similarly establish the gradient complexity of the proposed algorithm under these four different settings. [ABSTRACT FROM AUTHOR]

Multimarginal Optimal Transport (MOT) has attracted significant interest due to applications in machine learning, statistics, and the sciences. However, in most applications, the success of MOT is severely limited by a lack of efficient algorithms. Indeed, MOT in general requires exponential time in the number of marginals k and their support sizes n. This paper develops a general theory about what "structure" makes MOT solvable in poly (n , k) time. We develop a unified algorithmic framework for solving MOT in poly (n , k) time by characterizing the structure that different algorithms require in terms of simple variants of the dual feasibility oracle. This framework has several benefits. First, it enables us to show that the Sinkhorn algorithm, which is currently the most popular MOT algorithm, requires strictly more structure than other algorithms do to solve MOT in poly (n , k) time. Second, our framework makes it much simpler to develop poly (n , k) time algorithms for a given MOT problem. In particular, it is necessary and sufficient to (approximately) solve the dual feasibility oracle—which is much more amenable to standard algorithmic techniques. We illustrate this ease-of-use by developing poly (n , k) -time algorithms for three general classes of MOT cost structures: (1) graphical structure; (2) set-optimization structure; and (3) low-rank plus sparse structure. For structure (1), we recover the known result that Sinkhorn has poly (n , k) runtime; moreover, we provide the first poly (n , k) time algorithms for computing solutions that are exact and sparse. For structures (2)-(3), we give the first poly (n , k) time algorithms, even for approximate computation. Together, these three structures encompass many—if not most—current applications of MOT. [ABSTRACT FROM AUTHOR]

In this paper, we present difference of convex algorithms for solving bilevel programs in which the upper level objective functions are difference of convex functions, and the lower level programs are fully convex. This nontrivial class of bilevel programs provides a powerful modelling framework for dealing with applications arising from hyperparameter selection in machine learning. Thanks to the full convexity of the lower level program, the value function of the lower level program turns out to be convex and hence the bilevel program can be reformulated as a difference of convex bilevel program. We propose two algorithms for solving the reformulated difference of convex program and show their convergence to stationary points under very mild assumptions. Finally we conduct numerical experiments to a bilevel model of support vector machine classification. [ABSTRACT FROM AUTHOR]

Projection-free optimization via different variants of the Frank–Wolfe method has become one of the cornerstones of large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with self-concordance like properties. Such generalized self-concordant functions do not necessarily feature a Lipschitz continuous gradient, nor are they strongly convex, making them a challenging class of functions for first-order methods. Indeed, in a number of applications, such as inverse covariance estimation or distance-weighted discrimination problems in binary classification, the loss is given by a generalized self-concordant function having potentially unbounded curvature. For such problems projection-free minimization methods have no theoretical convergence guarantee. This paper closes this apparent gap in the literature by developing provably convergent Frank–Wolfe algorithms with standard O (1 / k) convergence rate guarantees. Based on these new insights, we show how these sublinearly convergent methods can be accelerated to yield linearly convergent projection-free methods, by either relying on the availability of a local liner minimization oracle, or a suitable modification of the away-step Frank–Wolfe method. [ABSTRACT FROM AUTHOR]