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In this paper, we address variational inequalities (VI) with a finite-sum structure. We introduce a novel single-loop stochastic variance-reduced algorithm, incorporating the Bregman distance function, and establish an optimal convergence guarantee under a monotone setting. Additionally, we explore a structured class of non-monotone problems that exhibit weak Minty solutions, and analyze the complexity of our proposed method, highlighting a significant improvement over existing approaches. Numerical experiments are presented to demonstrate the performance of our algorithm compared to state-of-the-art methods

In this paper, we explore a broad class of constrained saddle point problems with a bilevel structure, wherein the upper-level objective function is nonconvex-concave and smooth over compact and convex constraint sets, subject to a strongly convex lower-level objective function. This class of problems finds wide applicability in machine learning, encompassing robust multi-task learning, adversarial learning, and robust meta-learning. Our study extends the current literature in two main directions: (i) We consider a more general setting where the upper-level function is not necessarily strongly concave or linear in the maximization variable. (ii) While existing methods for solving saddle point problems with a bilevel structure are projected-based algorithms, we propose a one-sided projection-free method employing a linear minimization oracle. Specifically, by utilizing regularization and nested approximation techniques, we introduce a novel single-loop one-sided projection-free algorithm, requiring $\mathcal{O}(\epsilon^{-4})$ iterations to attain an $\epsilon$-stationary solution. Subsequently, we develop an efficient single-loop fully projected gradient-based algorithm capable of achieving an $\epsilon$-stationary solution within $\mathcal{O}(\epsilon^{-5}\log(1/\epsilon))$ iterations. When the upper-level objective function is linear in the maximization component, our results improve to $\mathcal{O}(\epsilon^{-3})$ and $\mathcal{O}(\epsilon^{-4})$, respectively. Finally, we tested our proposed methods against the state-of-the-art algorithms for solving a robust multi-task regression problem to showcase the superiority of our algorithms.

In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max\{1/\sqrt{\epsilon_{f}}, 1/\epsilon_g\})$ iterations to find a solution that is $\epsilon_f$-suboptimal and $\epsilon_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th H\"olderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max\{\epsilon_{f}^{-\frac{2r-1}{2r}},\epsilon_{g}^{-\frac{2r-1}{2r}}\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires $\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{2},1/\epsilon_g^{2}\}) $ stochastic oracle queries to obtain a solution that is $\epsilon_f$-optimal for the upper-level and $\epsilon_g$-optimal for the lower-level. This guarantee improves the previous best-known complexity of $\mathcal{O}(\max\{1/\epsilon_f^{4},1/\epsilon_g^{4}\})$. Moreover, for the case that the upper-level function is non-convex, our method requires at most $\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{3},1/\epsilon_g^{3}\}) $ stochastic oracle queries to find an $(\epsilon_f, \epsilon_g)$-stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon)$ and $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon^{2})$ for the convex and non-convex settings, respectively, where $\epsilon=\min \{\epsilon_f,\epsilon_g\}$.

In this paper, we investigate a class of constrained saddle point (SP) problems where the objective function is nonconvex-concave and smooth. This class of problems has wide applicability in machine learning, including robust multi-class classification and dictionary learning. Several projection-based primal-dual methods have been developed for tackling this problem; however, the availability of methods with projection-free oracles remains limited. To address this gap, we propose efficient single-loop projection-free methods reliant on first-order information. In particular, using regularization and nested approximation techniques, we propose a primal-dual conditional gradient method that solely employs linear minimization oracles to handle constraints. Assuming that the constraint set in the maximization is strongly convex, our method achieves an $\epsilon$-stationary solution within $\mathcal{O}(\epsilon^{-6})$ iterations. When the projection onto the constraint set of maximization is easy to compute, we propose a one-sided projection-free method that achieves an $\epsilon$-stationary solution within $\mathcal{O}(\epsilon^{-4})$ iterations. Moreover, we present improved iteration complexities of our methods under a strong concavity assumption. To the best of our knowledge, our proposed algorithms are among the first projection-free methods with convergence guarantees for solving nonconvex-concave SP problems., Comment: Accepted to NeurIPS 2023

Bilevel optimization is an important class of optimization problems where one optimization problem is nested within another. While various methods have emerged to address unconstrained general bilevel optimization problems, there has been a noticeable gap in research when it comes to methods tailored for the constrained scenario. The few methods that do accommodate constrained problems, often exhibit slow convergence rates or demand a high computational cost per iteration. To tackle this issue, our paper introduces a novel single-loop projection-free method employing a nested approximation technique. This innovative approach not only boasts an improved per-iteration complexity compared to existing methods but also achieves optimal convergence rate guarantees that match the best-known complexity of projection-free algorithms for solving convex constrained single-level optimization problems. In particular, when the hyper-objective function corresponding to the bilevel problem is convex, our method requires $\tilde{\mathcal{O}}(\epsilon^{-1})$ iterations to find an $\epsilon$-optimal solution. Moreover, when the hyper-objective function is non-convex, our method's complexity for finding an $\epsilon$-stationary point is $\mathcal{O}(\epsilon^{-2})$. To showcase the effectiveness of our approach, we present a series of numerical experiments that highlight its superior performance relative to state-of-the-art methods.

We consider a class of multi-agent cooperative consensus optimization problems with local nonlinear convex constraints where only those agents connected by an edge can directly communicate, hence, the optimal consensus decision lies in the intersection of these private sets. We develop an asynchronous distributed accelerated primal-dual algorithm to solve the considered problem. The proposed scheme is the first asynchronous method with an optimal convergence guarantee for this class of problems, to the best of our knowledge. In particular, we provide an optimal convergence rate of $\mathcal{O(1/K)}$ for suboptimality, infeasibility, and consensus violation.

Identifying, analyzing, and evaluating cybersecurity risks are essential to assess the vulnerabilities of modern manufacturing infrastructures and to devise effective decision-making strategies to secure critical manufacturing against potential cyberattacks. In response, this work proposes a graph-theoretic approach for risk modeling and assessment to address the lack of quantitative cybersecurity risk assessment frameworks for smart manufacturing systems. In doing so, first, threat attributes are represented using an attack graphical model derived from manufacturing cyberattack taxonomies. Attack taxonomies offer consistent structures to categorize threat attributes, and the graphical approach helps model their interdependence. Second, the graphs are analyzed to explore how threat events can propagate through the manufacturing value chain and identify the manufacturing assets that threat actors can access and compromise during a threat event. Third, the proposed method identifies the attack path that maximizes the likelihood of success and minimizes the attack detection probability, and then computes the associated cybersecurity risk. Finally, the proposed risk modeling and assessment framework is demonstrated via an interconnected smart manufacturing system illustrative example. Using the proposed approach, practitioners can identify critical connections and manufacturing assets requiring prioritized security controls and develop and deploy appropriate defense measures accordingly., Comment: 25 pages, 10 figures

In this paper, we study a class of bilevel optimization problems, also known as simple bilevel optimization, where we minimize a smooth objective function over the optimal solution set of another convex constrained optimization problem. Several iterative methods have been developed for tackling this class of problems. Alas, their convergence guarantees are either asymptotic for the upper-level objective, or the convergence rates are slow and sub-optimal. To address this issue, in this paper, we introduce a novel bilevel optimization method that locally approximates the solution set of the lower-level problem via a cutting plane, and then runs a conditional gradient update to decrease the upper-level objective. When the upper-level objective is convex, we show that our method requires ${\mathcal{O}}(\max\{1/\epsilon_f,1/\epsilon_g\})$ iterations to find a solution that is $\epsilon_f$-optimal for the upper-level objective and $\epsilon_g$-optimal for the lower-level objective. Moreover, when the upper-level objective is non-convex, our method requires ${\mathcal{O}}(\max\{1/\epsilon_f^2,1/(\epsilon_f\epsilon_g)\})$ iterations to find an $(\epsilon_f,\epsilon_g)$-optimal solution. We also prove stronger convergence guarantees under the H\"olderian error bound assumption on the lower-level problem. To the best of our knowledge, our method achieves the best-known iteration complexity for the considered class of bilevel problems., Comment: Accepted to AISTATS 2023

In this paper, we propose a variance-reduced primal-dual algorithm with Bregman distance for solving convex-concave saddle-point problems with finite-sum structure and nonbilinear coupling function. This type of problems typically arises in machine learning and game theory. Based on some standard assumptions, the algorithm is proved to converge with oracle complexity of $O\left(\frac{\sqrt n}{\epsilon}\right)$ and $O\left(\frac{n}{\sqrt \epsilon}+\frac{1}{\epsilon^{1.5}}\right)$ using constant and non-constant parameters, respectively where $n$ is the number of function components. Compared with existing methods, our framework yields a significant improvement over the number of required primal-dual gradient samples to achieve $\epsilon$-accuracy of the primal-dual gap. We tested our method for solving a distributionally robust optimization problem to show the effectiveness of the algorithm.

We propose decentralized primal-dual methods for cooperative multi-agent consensus optimization problems over both static and time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific convex functions over conic constraint sets defined by agent-specific nonlinear functions; hence, the optimal consensus decision should lie in the intersection of these private sets. Under the strong convexity assumption, we provide convergence rates for sub-optimality, infeasibility, and consensus violation in terms of the number of communications required; examine the effect of underlying network topology on the convergence rates., Comment: A preliminary result of this paper was presented in arXiv:1706.07907 by Hamedani and Aybat. In this paper, we generalize our results to the setting where agent-specific constraints are defined by nonlinear functions rather than linear ones which greatly improves the modeling capability. This generalization requires a more complicated analysis which is studied in this separate arXiv submission

We consider (stochastic) convex-concave saddle point (SP) problems with high-dimensional decision variables, arising in various machine learning problems. To contend with the challenges in computing full gradients, we employ a randomized block-coordinate primal-dual scheme in which randomly selected primal and dual blocks of variables are updated. We consider both deterministic and stochastic settings, where deterministic partial gradients and their randomly sampled estimates are used, respectively, at each iteration. We investigate the convergence of the proposed method under different blocking strategies and provide the corresponding complexity results. While the best-known complexity result for deterministic primal-dual methods using full gradients is $\mathcal O(\max\{M,N\}^2/\varepsilon)$ where $M$ and $N$ denote the number of primal and dual blocks, respectively, we show that our proposed randomized block-coordinate method can achieve an improved convergence rate of $\mathcal O(MN/\varepsilon)$. Moreover, for the stochastic setting where a mini-batch sample gradient is utilized, we show an optimal oracle complexity of $\tilde{\mathcal{O}}(M^2N^2/\varepsilon^2)$ through acceleration. Finally, almost sure convergence of the iterate sequence to a saddle point is established., Comment: This is a major update. Complexity results are established under stochastic first-order oracle setting rather than the finite-sum form considered in earlier versions. All the theorems and their proofs are revised. We discuss when blocking both the primal and dual vector leads to better complexity bounds than the complexity without using blocks or with blocking either the primal or dual alone

In this paper, we propose a primal-dual algorithm with a novel momentum term using the partial gradients of the coupling function that can be viewed as a generalization of the method proposed by Chambolle and Pock in 2016 to solve saddle point problems defined by a convex-concave function $\mathcal L(x,y)=f(x)+\Phi(x,y)-h(y)$ with a general coupling term $\Phi(x,y)$ that is not assumed to be bilinear. Assuming $\nabla_x\Phi(\cdot,y)$ is Lipschitz for any fixed $y$, and $\nabla_y\Phi(\cdot,\cdot)$ is Lipschitz, we show that the iterate sequence converges to a saddle point; and for any $(x,y)$, we derive error bounds in terms of $\mathcal L(\bar{x}_k,y)-\mathcal L(x,\bar{y}_k)$ for the ergodic sequence $\{\bar{x}_k,\bar{y}_k\}$. In particular, we show $\mathcal O(1/k)$ rate when the problem is merely convex in $x$. Furthermore, assuming $\Phi(x,\cdot)$ is linear for each fixed $x$ and $f$ is strongly convex, we obtain the ergodic convergence rate of $\mathcal O(1/k^2)$ -- we are not aware of another single-loop method in the related literature achieving the same rate when $\Phi$ is not bilinear. Finally, we propose a backtracking technique which does not require the knowledge of Lipschitz constants while ensuring the same convergence results. We also consider convex optimization problems with nonlinear functional constraints and we show that using the backtracking scheme, the optimal convergence rate can be achieved even when the dual domain is unbounded. We tested our method against other state-of-the-art first-order algorithms and interior-point methods for solving quadratically constrained quadratic problems with synthetic data, the kernel matrix learning, and regression with fairness constraints arising in machine learning., Comment: linesearch is added; new numerical experiment is added; important remarks are added in this version

We consider cooperative multi-agent consensus optimization problems over both static and time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific possibly non-smooth composite convex functions over agent-specific private conic constraint sets; hence, the optimal consensus decision should lie in the intersection of these private sets. Assuming the sum function is strongly convex, we provide convergence rates in suboptimality, infeasibility and consensus violation; examine the effect of underlying network topology on the convergence rates of the proposed decentralized algorithms.

We consider cooperative multi-agent resource sharing problems over time-varying communication networks, where only local communications are allowed. The objective is to minimize the sum of agent-specific composite convex functions subject to a conic constraint that couples agents' decisions. We propose a distributed primal-dual algorithm DPDA-D to solve the saddle point formulation of the sharing problem on time-varying (un)directed communication networks; and we show that primal-dual iterate sequence converges to a point defined by a primal optimal solution and a consensual dual price for the coupling constraint. Furthermore, we provide convergence rates for suboptimality, infeasibility and consensus violation of agents' dual price assessments; examine the effect of underlying network topology on the convergence rates of the proposed decentralized algorithm; and compare DPDA-D with a centralized method on the basis pursuit denoising and multi-channel power allocation problems.

We consider cooperative multi-agent consensus optimization problems over an undirected network of agents, where only those agents connected by an edge can directly communicate. The objective is to minimize the sum of agent-specific composite convex functions over agent-specific private conic constraint sets; hence, the optimal consensus decision should lie in the intersection of these private sets. We provide convergence rates both in sub-optimality, infeasibility and consensus violation; examine the effect of underlying network topology on the convergence rates of the proposed decentralized algorithms; and show how to extend these methods to handle time-varying communications networks and to solve problems with resource sharing constraints.

Many research has been conducted about quadratic programming and inverse optimization. In this paper we present the combination aspect of these subjects, applying on transportation problem. First, we obtain the inverse form of quadratic tranportation problem under $L_1$ norm by using duality as well as introducing the optimal value. Then, we do the same process for inverse quadratic transportation problem (IQTP) under $L_\infty$ norm.