Convex-concave Saddle Point Problems Research Articles

Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem

AbstractThis paper studies the convergence rate of a second-order dynamical system associated with a nonsmooth bilinearly coupled convex-concave saddle point problem, as well as the convergence rate of its corresponding discretizations. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on an implicit discretization scheme, we propose a primal-dual algorithm and provide a non-ergodic convergence rate under a general setting for the inertial parameters when one objective function is continuously differentiable and convex and the other is a proper, convex and lower semicontinuous function. For this algorithm we derive a $$O\left( 1/k^2 \right) $$ O 1 / k 2 convergence rate under three classical rules proposed by Nesterov, Chambolle-Dossal and Attouch-Cabot without assuming strong convexity, which is compatible with the results of the continuous-time dynamic system. For the case when both objective functions are continuously differentiable and convex, we further present a primal-dual algorithm based on an explicit discretization. We provide a corresponding non-ergodic convergence rate for this algorithm and show that the sequence of iterates generated weakly converges to a primal-dual optimal solution. Finally, we present numerical experiments that indicate the superior numerical performance of both algorithms.

Non-ergodic convergence rate of an inertial accelerated primal–dual algorithm for saddle point problems

In this paper, we design an inertial accelerated primal–dual algorithm to address the convex–concave saddle point problem, which is formulated as minxmaxyf(x)+〈Kx,y〉−g(y). Remarkably, both functions f and g exhibit a composite structure, combining “nonsmooth” ＋ “smooth” components. Under the assumption of partially strong convexity in the sense that f is convex and g is strongly convex, we introduce a novel inertial accelerated primal–dual algorithm based on Nesterov’s extrapolation. This algorithm can be reduced to two classical accelerated forward–backward methods for unconstrained optimization problem. We show that the proposed algorithm achieves a non-ergodic O(1/k2) convergence rate, where k represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm.

Convergence rate analysis of the gradient descent–ascent method for convex–concave saddle-point problems

In this paper, we study the gradient descent–ascent method for convex–concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming performance estimation method. The given convergence rate incorporates most parameters of the problem and it is exact for a large class of strongly convex-strongly concave saddle-point problems for one iteration. We also investigate the algorithm without strong convexity and we provide some necessary and sufficient conditions under which the gradient descent–ascent enjoys linear convergence.

Decentralized saddle point problems via non-Euclidean mirror prox

We consider smooth convex-concave saddle point problems in the decentralized distributed setting, where a finite-sum objective is distributed among the nodes of a computational network. At each node, the local objective depends on the groups of local and global variables. For such problems, we propose a decentralized distributed algorithm with O ( ϵ − 1 ) communication and oracle calls complexities to achieve accuracy ε in terms of the duality gap and in terms of consensus between nodes. Further, we prove lower bounds for the communication and oracle calls complexities and show that our algorithm matches these bounds, i.e. it is optimal. In contrast to existing decentralized algorithms, our algorithm admits non-euclidean proximal setup, including, e.g. entropic. We illustrate the work of the proposed algorithm on the prominent problem of computing Wasserstein barycenters (WB), where a non-euclidean proximal setup arises naturally in a bilinear saddle point reformulation of the WB problem.

A Primal-Dual Smoothing Framework for Max-Structured Non-Convex Optimization

We propose a primal-dual smoothing framework for finding a near-stationary point of a class of nonsmooth nonconvex optimization problems with max-structure. We analyze the primal and dual gradient complexities of the framework via two approaches, that is, the dual-then-primal and primal-the-dual smoothing approaches. Our framework improves the best-known oracle complexities of the existing method, even in the restricted problem setting. As an important part of our framework, we propose a first-order method for solving a class of (strongly) convex-concave saddle-point problems, which is based on a newly developed non-Hilbertian inexact accelerated proximal gradient algorithm for strongly convex composite minimization that enjoys duality-gap convergence guarantees. Some variants and extensions of our framework are also discussed. Funding: R. Zhao’s research is partially supported by AFOSR [Grant FA9550-22-1-0356].

Primal-Dual First-Order Methods for Affinely Constrained Multi-block Saddle Point Problems

We consider the convex-concave saddle point problem , where the decision variables and/or are subject to certain multi-block structure and affine coupling constraints, and possesses certain separable structure. Although the minimization counterpart of this problem has been widely studied under the topics of ADMM, this minimax problem is rarely investigated. In this paper, a convenient notion of -saddle point is proposed, under which the convergence rate of several proposed algorithms are analyzed. When only one of and has multiple blocks and affine constraint, several natural extensions of ADMM are proposed to solve the problem. Depending on the number of blocks and the level of smoothness, or convergence rates are derived for our algorithms. When both and have multiple blocks and affine constraints, a new algorithm called xtra- radient Method of ultipliers (EGMM) is proposed. Under desirable smoothness conditions, an rate of convergence can be guaranteed regardless of the number of blocks in and . An in-depth comparison between EGMM (fully primal-dual method) and ADMM (approximate dual method) is made over the multi-block optimization problems to illustrate the advantage of the EGMM.

Solving Optimization Problems with Blackwell Approachability

In this paper, we propose a new algorithm for solving convex-concave saddle-point problems using regret minimization in the repeated game framework. To do so, we introduce the Conic Blackwell Algorithm+ ([Formula: see text]), a new parameter- and scale-free regret minimizer for general convex compact sets. [Formula: see text] is based on Blackwell approachability and attains [Formula: see text] regret. We show how to efficiently instantiate [Formula: see text] for many decision sets of interest, including the simplex, [Formula: see text] norm balls, and ellipsoidal confidence regions in the simplex. Based on [Formula: see text], we introduce [Formula: see text], a new parameter-free algorithm for solving convex-concave saddle-point problems achieving a [Formula: see text] ergodic convergence rate. In our simulations, we demonstrate the wide applicability of [Formula: see text] on several standard saddle-point problems from the optimization and operations research literature, including matrix games, extensive-form games, distributionally robust logistic regression, and Markov decision processes. In each setting, [Formula: see text] achieves state-of-the-art numerical performance and outperforms classical methods, without the need for any choice of step sizes or other algorithmic parameters. Funding: J. Grand-Clément is supported by the Agence Nationale de la Recherche [Grant 11-LABX-0047] and by Hi! Paris. C. Kroer is supported by the Office of Naval Research [Grant N00014-22-1-2530] and by the National Science Foundation [Grant IIS-2147361].

A Unified Analysis of Variational Inequality Methods: Variance Reduction, Sampling, Quantization, and Coordinate Descent

In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient method (the classic algorithm for variational inequalities) and consider the strongly monotone and monotone cases, which corresponds to strongly-convex-strongly-concave and convex-concave saddle point problems. The theoretical analysis is based on parametric assumptions about Extra-Gradient iterations. Therefore, it can serve as a strong basis for combining the already existing type methods and also for creating new algorithms. In particular, to show this we develop new robust methods, which include methods with quantization, coordinate methods, distributed randomized local methods, and others. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is also confirmed by numerical experiments with GANs.

Chambolle–Pock’s Primal-Dual Method with Mismatched Adjoint

The primal-dual method of Chambolle and Pock is a widely used algorithm to solve various optimization problems written as convex-concave saddle point problems. Each update step involves the application of both the forward linear operator and its adjoint. However, in practical applications like computerized tomography, it is often computationally favourable to replace the adjoint operator by a computationally more efficient approximation. This leads to an adjoint mismatch in the algorithm. In this paper, we analyze the convergence of Chambolle–Pock’s primal-dual method under the presence of a mismatched adjoint in the strongly convex setting. We present an upper bound on the error of the primal solution and derive stepsizes and mild conditions under which convergence to a fixed point is still guaranteed. Furthermore we show linear convergence similar to the result of Chambolle–Pock’s primal-dual method without the adjoint mismatch. Moreover, we illustrate our results both for an academic and a real-world inspired application.

Solving strongly convex-concave composite saddle-point problems with low dimension of one group of variable

Algorithmic methods are developed that guarantee efficient complexity estimates for strongly convex-concave saddle-point problems in the case when one group of variables has a high dimension, while another has a rather low dimension (up to 100). These methods are based on reducing problems of this type to the minimization (maximization) problem for a convex (concave) functional with respect to one of the variables such that an approximate value of the gradient at an arbitrary point can be obtained with the required accuracy using an auxiliary optimization subproblem with respect to the other variable. It is proposed to use the ellipsoid method and Vaidya's method for low-dimensional problems and accelerated gradient methods with inexact information about the gradient or subgradient for high-dimensional problems. In the case when one group of variables, ranging over a hypercube, has a very low dimension (up to five), another proposed approach to strongly convex-concave saddle-point problems is rather efficient. This approach is based on a new version of a multidimensional analogue of Nesterov's method on a square (the multidimensional dichotomy method) with the possibility to use inexact values of the gradient of the objective functional. Bibliography: 28 titles.

Adaptive and Universal Algorithms for Variational Inequalities with Optimal Convergence

We develop new adaptive algorithms for variational inequalities with monotone operators, which capture many problems of interest, notably convex optimization and convex-concave saddle point problems. Our algorithms automatically adapt to unknown problem parameters such as the smoothness and the norm of the operator, and the variance of the stochastic evaluation oracle. We show that our algorithms are universal and simultaneously achieve the optimal convergence rates in the non-smooth, smooth, and stochastic settings. The convergence guarantees of our algorithms improve over existing adaptive methods and match the optimal non-adaptive algorithms. Additionally, prior works require that the optimization domain is bounded. In this work, we remove this restriction and give algorithms for unbounded domains that are adaptive and universal. Our general proof techniques can be used for many variants of the algorithm using one or two operator evaluations per iteration. The classical methods based on the ExtraGradient/MirrorProx algorithm require two operator evaluations per iteration, which is the dominant factor in the running time in many settings.

Subgradient ellipsoid method for nonsmooth convex problems

In this paper, we present a new ellipsoid-type algorithm for solving nonsmooth problems with convex structure. Examples of such problems include nonsmooth convex minimization problems, convex-concave saddle-point problems and variational inequalities with monotone operator. Our algorithm can be seen as a combination of the standard Subgradient and Ellipsoid methods. However, in contrast to the latter one, the proposed method has a reasonable convergence rate even when the dimensionality of the problem is sufficiently large. For generating accuracy certificates in our algorithm, we propose an efficient technique, which ameliorates the previously known recipes (Nemirovski in Math Oper Res 35(1):52–78, 2010).

An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function

In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a proximal step in the nonsmooth component of the coupling function. We consider the situations convex-concave, convex-strongly concave and strongly convex-strongly concave related to the saddle point problem under investigation. Regarding iterates we obtain (weak) convergence, a convergence rate of order mathcal {O}(frac{1}{K}) and linear convergence like mathcal {O}(theta ^{K}) with theta < 1, respectively. In terms of function values we obtain ergodic convergence rates of order mathcal {O}(frac{1}{K}), mathcal {O}(frac{1}{K^{2}}) and mathcal {O}(theta ^{K}) with theta < 1, respectively. We validate our theoretical considerations on a nonsmooth-linear saddle point problem, the training of multi kernel support vector machines and a classification problem incorporating minimax group fairness.

Extragradient and extrapolation methods with generalized Bregman distances for saddle point problems

In this paper, we introduce two Bregman-type algorithmic frameworks to generalize the extragradient and extrapolation methods. With the help of relative Lipschitzness and the Bregman distance tool, the iteration properties of the proposed frameworks are analyzed. As applied to smooth convex-concave saddle point problems, our theory rediscovers the main results in Mokhtari et al. (2020) [14] for wider frameworks under weaker assumptions via a conceptually different approach.

Accelerated Stochastic Algorithms for Convex-Concave Saddle-Point Problems

We develop stochastic first-order primal-dual algorithms to solve a class of convex-concave saddle-point problems. When the saddle function is strongly convex in the primal variable, we develop the first stochastic restart scheme for this problem. When the gradient noises obey sub-Gaussian distributions, the oracle complexity of our restart scheme is strictly better than any of the existing methods, even in the deterministic case. Furthermore, for each problem parameter of interest, whenever the lower bound exists, the oracle complexity of our restart scheme is either optimal or nearly optimal (up to a log factor). The subroutine used in this scheme is itself a new stochastic algorithm developed for the problem where the saddle function is nonstrongly convex in the primal variable. This new algorithm, which is based on the primal-dual hybrid gradient framework, achieves the state-of-the-art oracle complexity and may be of independent interest.

Over-the-Air Computation via Reconfigurable Intelligent Surface

Over-the-air computation (AirComp) is a disruptive technique for fast wireless data aggregation in Internet of Things (IoT) networks via exploiting the waveform superposition property of multiple-access channels. However, the performance of AirComp is bottlenecked by the worst channel condition among all links between the IoT devices and the access point. In this paper, a reconfigurable intelligent surface (RIS) assisted AirComp system is proposed to boost the received signal power and thus mitigate the performance bottleneck by reconfiguring the propagation channels. With an objective to minimize the AirComp distortion, we propose a joint design of AirComp transceivers and RIS phase-shifts, which however turns out to be a highly intractable non-convex programming problem. To this end, we develop a novel alternating minimization framework in conjunction with the successive convex approximation technique, which is proved to converge monotonically. To reduce the computational complexity, we transform the subproblem in each alternation as a smooth convex-concave saddle point problem, which is then tackled by proposing a Mirror-Prox method that only involves a sequence of closed-form updates. Simulations show that the computation time of the proposed algorithm can be two orders of magnitude smaller than that of the state-of-the-art algorithms, while achieving a similar distortion performance.

Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games

We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean-field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are not approachable with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex-concave saddle-point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems.

On well-structured convex–concave saddle point problems and variational inequalities with monotone operators

ABSTRACT For those acquainted with CVX (aka disciplined convex programming) of Grant and Boyd (Matlab software for disciplined convex programming, version 2.2, CVX Research, Inc., 2020. http://cvxr.com/cvx/doc/), the motivation of this work is the desire to extend the scope of CVX beyond convex minimization – to convex–concave saddle point problems and variational inequalities with monotone operators. To attain this goal, given a family of cones (e.g. Lorentz, semidefinite, geometric, etc.), we introduce the notions of -conic representation of a convex–concave saddle point problem and of variational inequality with monotone operator. We demonstrate that given such a representation of the problem of interest, the latter can be reduced straightforwardly to a conic problem on a cone from and thus can be solved by (any) solver capable to handle conic problems on cones from (e.g. Mosek or SDPT3 in the case of semidefinite cones). We also show that -representations of convex–concave functions and monotone vector fields admit a fully algorithmic calculus which helps to recognize the cases when a saddle point problem or variational inequality can be converted into a conic problem on a cone from and to carry out such conversion.

On lower iteration complexity bounds for the convex concave saddle point problems

In this paper, we study the lower iteration complexity bounds for finding the saddle point of a strongly convex and strongly concave saddle point problem: $\min_x\max_yF(x,y)$. We restrict the classes of algorithms in our investigation to be either pure first-order methods or methods using proximal mappings. The existing lower bound result for this type of problems is obtained via the framework of strongly monotone variational inequality problems, which corresponds to the case where the gradient Lipschitz constants ($L_x, L_y$ and $L_{xy}$) and strong convexity/concavity constants ($\mu_x$ and $\mu_y$) are uniform with respect to variables $x$ and $y$. However, specific to the min-max saddle point problem these parameters are naturally different. Therefore, one is led to finding the best possible lower iteration complexity bounds, specific to the min-max saddle point models. In this paper we present the following results. For the class of pure first-order algorithms, our lower iteration complexity bound is $\Omega\left(\sqrt{\frac{L_x}{\mu_x}+\frac{L_{xy}^2}{\mu_x\mu_y}+\frac{L_y}{\mu_y}}\cdot\ln\left(\frac{1}{\epsilon}\right)\right)$, where the term $\frac{L_{xy}^2}{\mu_x\mu_y}$ explains how the coupling influences the iteration complexity. Under several special parameter regimes, this lower bound has been achieved by corresponding optimal algorithms. However, whether or not the bound under the general parameter regime is optimal remains open. Additionally, for the special case of bilinear coupling problems, given the availability of certain proximal operators, a lower bound of $\Omega\left(\sqrt{\frac{L_{xy}^2}{\mu_x\mu_y}+1}\cdot\ln(\frac{1}{\epsilon})\right)$ is established in this paper, and optimal algorithms have already been developed in the literature.

A first-order inexact primal-dual algorithm for a class of convex-concave saddle point problems

In this paper, we study a first-order inexact primal-dual algorithm (I-PDA) for solving a class of convex-concave saddle point problems. The I-PDA, which involves a relative error criterion and generalizes the classical PDA, has the advantage of solving one subproblem inexactly when it does not have a closed-form solution. We show that the whole sequence generated by I-PDA converges to a saddle point solution with $\mathcal {O}(1/N)$ ergodic convergence rate, where N is the iteration number. In addition, under a mild calmness condition, we establish the global Q-linear convergence rate of the distance between the iterates generated by I-PDA and the solution set, and the R-linear convergence speed of the nonergodic iterates. Furthermore, we demonstrate that many problems arising from practical applications satisfy this calmness condition. Finally, some numerical experiments are performed to show the superiority and linear convergence behaviors of I-PDA.