arXiv subjects

Farzad Yousefian

Publications and source records attributed to Farzad Yousefian.

At least 37 records · Page 2Linked to original sources

arXiv2021

Zeroth-order randomized block methods for constrained minimization of expectation-valued Lipschitz continuous functions

We consider the minimization of an $L_0$-Lipschitz continuous and expectation-valued function, denoted by $f$ and defined as $f(x)\triangleq \mathbb{E}[\tilde{f}(x,ω)]$, over a Cartesian product of closed and convex sets with a view towards obtaining both asymptotics as well as rate and complexity guarantees for computing an approximate stationary point (in a Clarke sense). We adopt a smoothing-based approach reliant on minimizing $f_η$ where $f_η(x) \triangleq \mathbb{E}_{u}[f(x+ηu)]$, $u$ is a random variable defined on a unit sphere, and $η> 0$. In fact, it is observed that a stationary point of the $η$-smoothed problem is a $2η$-stationary point for the original problem in the Clarke sense. In such a setting, we derive a suitable residual function that provides a metric for stationarity for the smoothed problem. By leveraging a zeroth-order framework reliant on utilizing sampled function evaluations implemented in a block-structured regime, we make two sets of contributions for the sequence generated by the proposed scheme. (i) The residual function of the smoothed problem tends to zero almost surely along the generated sequence; (ii) To compute an $x$ that ensures that the expected norm of the residual of the $η$-smoothed problem is within $ε$ requires no greater than $\mathcal{O}(\tfrac{1}{ηε^2})$ projection steps and $\mathcal{O}\left(\tfrac{1}{η^2 ε^4}\right)$ function evaluations. These statements appear to be novel and there appear to be few results to contend with general nonsmooth, nonconvex, and stochastic regimes via zeroth-order approaches.

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arXiv2021

Bilevel Distributed Optimization in Directed Networks

Motivated by emerging applications in wireless sensor networks and large-scale data processing, we consider distributed optimization over directed networks where the agents communicate their information locally to their neighbors to cooperatively minimize a global cost function. We introduce a new unifying distributed constrained optimization model that is characterized as a bilevel optimization problem. This model captures a wide range of existing problems over directed networks including: (i) Distributed optimization with linear constraints; (ii) Distributed unconstrained nonstrongly convex optimization over directed networks. Employing a novel regularization-based relaxation approach and gradient-tracking schemes, we develop an iteratively regularized push-pull gradient algorithm. We establish the consensus and derive new convergence rate statements for suboptimality and infeasibility of the generated iterates for solving the bilevel model. The proposed algorithm and the complexity analysis obtained in this work appear to be new for addressing the bilevel model and also for the two sub-classes of problems. The numerical performance of the proposed algorithm is presented.

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arXiv2021

An Incremental Gradient Method for Large-scale Distributed Nonlinearly Constrained Optimization

Motivated by applications arising from sensor networks and machine learning, we consider the problem of minimizing a finite sum of nondifferentiable convex functions where each component function is associated with an agent and a hard-to-project constraint set. Among well-known avenues to address finite sum problems is the class of incremental gradient (IG) methods where a single component function is selected at each iteration in a cyclic or randomized manner. When the problem is constrained, the existing IG schemes (including projected IG, proximal IAG, and SAGA) require a projection step onto the feasible set at each iteration. Consequently, the performance of these schemes is afflicted with costly projections when the problem includes: (1) nonlinear constraints, or (2) a large number of linear constraints. Our focus in this paper lies in addressing both of these challenges. We develop an algorithm called averaged iteratively regularized incremental gradient (aIR-IG) that does not involve any hard-to-project computation. Under mild assumptions, we derive non-asymptotic rates of convergence for both suboptimality and infeasibility metrics. Numerically, we show that the proposed scheme outperforms the standard projected IG methods on distributed soft-margin support vector machine problems.

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arXiv2021

A Method with Convergence Rates for Optimization Problems with Variational Inequality Constraints

We consider a class of optimization problems with Cartesian variational inequality (CVI) constraints, where the objective function is convex and the CVI is associated with a monotone mapping and a convex Cartesian product set. This mathematical formulation captures a wide range of optimization problems including those complicated by the presence of equilibrium constraints, complementarity constraints, or an inner-level large scale optimization problem. In particular, an important motivating application arises from the notion of efficiency estimation of equilibria in multi-agent networks, e.g., communication networks and power systems. In the literature, the iteration complexity of the existing solution methods for optimization problems with CVI constraints appears to be unknown. To address this shortcoming, we develop a first-order method called averaging randomized block iteratively regularized gradient (aRB-IRG). The main contributions include: (i) In the case where the associated set of the CVI is bounded and the objective function is nondifferentiable and convex, we derive new non-asymptotic suboptimality and infeasibility convergence rate statements in a mean sense. We also obtain deterministic variants of the convergence rates when we suppress the randomized block-coordinate scheme. Importantly, this paper appears to be the first work to provide these rate guarantees for this class of problems. (ii) In the case where the CVI set is unbounded and the objective function is smooth and strongly convex, utilizing the properties of the Tikhonov trajectory, we establish the global convergence of aRB-IRG in an almost sure and a mean sense. We provide the numerical experiments for computing the best Nash equilibrium in a networked Cournot competition model.

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arXiv2020

A Variable Sample-size Stochastic Quasi-Newton Method for Smooth and Nonsmooth Stochastic Convex Optimization

Classical theory for quasi-Newton schemes has focused on smooth deterministic unconstrained optimization while recent forays into stochastic convex optimization have largely resided in smooth, unconstrained, and strongly convex regimes. Naturally, there is a compelling need to address nonsmoothness, the lack of strong convexity, and the presence of constraints. Accordingly, this paper presents a quasi-Newton framework that can process merely convex and possibly nonsmooth (but smoothable) stochastic convex problems. We propose a framework that combines iterative smoothing and regularization with a variance-reduced scheme reliant on using increasing sample-sizes of gradients. We make the following contributions. (i) We develop a regularized and smoothed variable sample-size BFGS update (rsL-BFGS) that generates a sequence of Hessian approximations and can accommodate nonsmooth convex objectives by utilizing iterative regularization and smoothing. (ii) In strongly convex regimes with state-dependent noise, the proposed variable sample-size stochastic quasi-Newton scheme admits a non-asymptotic linear rate of convergence while the oracle complexity of computing an $ε$-solution is $\mathcal{O}(κ^{m+1}/ε)$ where $κ$ is the condition number and $m\geq 1$. In nonsmooth (but smoothable) regimes, using Moreau smoothing retains the linear convergence rate. To contend with the possible unavailability of Lipschitzian and strong convexity parameters, we also provide sublinear rates; (iii) In merely convex but smooth settings, the regularized VS-SQN scheme rVS-SQN displays a rate of $\mathcal{O}(1/k^{(1-\varepsilon)})$. When the smoothness requirements are weakened, the rate for the regularized and smoothed VS-SQN scheme worsens to $\mathcal{O}(k^{-1/3})$. Such statements allow for a state-dependent noise assumption under a quadratic growth property.

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arXiv2019

On stochastic and deterministic quasi-Newton methods for non-Strongly convex optimization: Asymptotic convergence and rate analysis

Motivated by applications arising from large scale optimization and machine learning, we consider stochastic quasi-Newton (SQN) methods for solving unconstrained convex optimization problems. The convergence analysis of the SQN methods, both full and limited-memory variants, require the objective function to be strongly convex. However, this assumption is fairly restrictive and does not hold for applications such as minimizing the logistic regression loss function. To the best of our knowledge, no rate statements currently exist for SQN methods in the absence of such an assumption. Also, among the existing first-order methods for addressing stochastic optimization problems with merely convex objectives, those equipped with provable convergence rates employ averaging. However, this averaging technique has a detrimental impact on inducing sparsity. Motivated by these gaps, the main contributions of the paper are as follows: (i) Addressing large scale stochastic optimization problems, we develop an iteratively regularized stochastic limited-memory BFGS (IRS-LBFGS) algorithm, where the stepsize, regularization parameter, and the Hessian inverse approximation matrix are updated iteratively. We establish the convergence to an optimal solution of the original problem both in an almost-sure and mean senses. We derive the convergence rate in terms of the objective function's values and show that it is of the order $\mathcal{O}\left(k^{-\left(\frac{1}{3}-ε\right)}\right)$, where $ε$ is an arbitrary small positive scalar; (ii) In deterministic regime, we show that the regularized limited-memory BFGS algorithm displays a rate of the order $\mathcal{O}\left(\frac{1}{k^{1 -ε'}}\right)$, where $ε'$ is an arbitrary small positive scalar. We present our numerical experiments performed on a large scale text classification problem.

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arXiv2019

An iterative regularized mirror descent method for ill-posed nondifferentiable stochastic optimization

A wide range of applications arising in machine learning and signal processing can be cast as convex optimization problems. These problems are often ill-posed, i.e., the optimal solution lacks a desired property such as uniqueness or sparsity. In the literature, to address ill-posedness, a bilevel optimization problem is considered where the goal is to find among optimal solutions of the inner level optimization problem, a solution that minimizes a secondary metric, i.e., the outer level objective function. In addressing the resulting bilevel model, the convergence analysis of most existing methods is limited to the case where both inner and outer level objectives are differentiable deterministic functions. While these assumptions may not hold in big data applications, to the best of our knowledge, no solution method equipped with complexity analysis exists to address presence of uncertainty and nondifferentiability in both levels in this class of problems. Motivated by this gap, we develop a first-order method called Iterative Regularized Stochastic Mirror Descent (IR-SMD). We establish the global convergence of the iterate generated by the algorithm to the optimal solution of the bilevel problem in an almost sure and a mean sense. We derive a convergence rate of ${\cal O}\left(1/N^{0.5-δ}\right)$ for the inner level problem, where $δ>0$ is an arbitrary small scalar. Numerical experiments for solving two classes of bilevel problems, including a large scale binary text classification application, are presented.

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arXiv2019

First-order Methods with Convergence Rates for Multi-agent Systems on Semidefinite Matrix Spaces

The goal in this paper is to develop first-order methods equipped with convergence rates for multi-agent optimization problems on semidefinite matrix spaces. These problems include cooperative optimization problems and non-cooperative Nash games. Accordingly, first we consider a multi-agent system where the agents cooperatively minimize the summation of their local convex objectives, and second, we consider Cartesian stochastic variational inequality (CSVI) problems with monotone mappings for addressing stochastic Nash games on semidefinite matrix spaces. Despite the recent advancements in first-order methods addressing problems over vector spaces, there seems to be a major gap in the theory of the first-order methods for optimization problems and equilibriums on semidefinite matrix spaces. In particular, to the best of our knowledge, there exists no method with provable convergence rate for solving the two classes of problems under mild assumptions. Most existing methods either rely on strong assumptions, or require a two-loop framework where at each iteration a semidefinite optimization problem needs to be solved. Motivated by this gap, in the first part of the paper, we develop a mirror descent incremental subgradient method for minimizing a finite-sum function. We show that the iterates generated by the algorithm converge asymptotically to an optimal solution and derive a non-asymptotic convergence rate. In the second part, we consider semidefinite CSVI problems. We develop a stochastic mirror descent method that only requires monotonicity of the mapping. We show that the iterates generated by the algorithm converge to a solution of the CSVI almost surely. Using a suitably defined gap function, we derive a convergence rate statement. This work appears to be the first that provides a convergence speed guarantee for monotone CSVIs on semidefinite matrix spaces.

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arXiv2018

A Randomized Block Coordinate Iterative Regularized Gradient Method for High-dimensional Ill-posed Convex Optimization

Motivated by high-dimensional nonlinear optimization problems as well as ill-posed optimization problems arising in image processing, we consider a bilevel optimization model where we seek among the optimal solutions of the inner level problem, a solution that minimizes a secondary metric. Our goal is to address the high-dimensionality of the bilevel problem, and the nondifferentiability of the objective function. Minimal norm gradient, sequential averaging, and iterative regularization are some of the recent schemes developed for addressing the bilevel problem. But none of them address the high-dimensional structure and nondifferentiability. With this gap in the literature, we develop a randomized block coordinate iterative regularized gradient descent scheme (RB-IRG). We establish the convergence of the sequence generated by RB-IRG to the unique solution of the bilevel problem of interest. Furthermore, we derive a rate of convergence $\mathcal{O} \left(\frac{1}{{k}^{0.5-δ}}\right)$, with respect to the inner level objective function. We demonstrate the performance of RB-IRG in solving the ill-posed problems arising in image processing.

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arXiv2018

An Iterative Regularized Incremental Projected Subgradient Method for a Class of Bilevel Optimization Problems

We study a class of bilevel convex optimization problems where the goal is to find the minimizer of an objective function in the upper level, among the set of all optimal solutions of an optimization problem in the lower level. A wide range of problems in convex optimization can be formulated using this class. An important example is the case where an optimization problem is ill-posed. In this paper, our interest lies in addressing the bilevel problems, where the lower level objective is given as a finite sum of separate nondifferentiable convex component functions. This is the case in a variety of applications in distributed optimization, such as large-scale data processing in machine learning and neural networks. To the best of our knowledge, this class of bilevel problems, with a finite sum in the lower level, has not been addressed before. Motivated by this gap, we develop an iterative regularized incremental subgradient method, where the agents update their iterates in a cyclic manner using a regularized subgradient. Under a suitable choice of the regularization parameter sequence, we establish the convergence of the proposed algorithm and derive a rate of $\mathcal{O} \left({1}/k^{0.5-ε}\right)$ in terms of the lower level objective function for an arbitrary small $ε>0$. We present the performance of the algorithm on a binary text classification problem.

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arXiv2018

A First-order Method for Monotone Stochastic Variational Inequalities on Semidefinite Matrix Spaces

Motivated by multi-user optimization problems and non-cooperative Nash games in stochastic regimes, we consider stochastic variational inequality (SVI) problems on matrix spaces where the variables are positive semidefinite matrices and the mapping is merely monotone. Much of the interest in the theory of variational inequality (VI) has focused on addressing VIs on vector spaces.Yet, most existing methods either rely on strong assumptions, or require a two-loop framework where at each iteration, a projection problem, i.e., a semidefinite optimization problem needs to be solved. Motivated by this gap, we develop a stochastic mirror descent method where we choose the distance generating function to be defined as the quantum entropy. This method is a single-loop first-order method in the sense that it only requires a gradient-type of update at each iteration. The novelty of this work lies in the convergence analysis that is carried out through employing an auxiliary sequence of stochastic matrices. Our contribution is three-fold: (i) under this setting and employing averaging techniques, we show that the iterate generated by the algorithm converges to a weak solution of the SVI; (ii) moreover, we derive a convergence rate in terms of the expected value of a suitably defined gap function; (iii) we implement the developed method for solving a multiple-input multiple-output multi-cell cellular wireless network composed of seven hexagonal cells and present the numerical experiments supporting the convergence of the proposed method.

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arXiv2018

Self-tuned mirror descent schemes for smooth and nonsmooth high-dimensional stochastic optimization

We consider randomized block coordinate stochastic mirror descent (RBSMD) methods for solving high-dimensional stochastic optimization problems with strongly convex objective functions. Our goal is to develop RBSMD schemes that achieve a rate of convergence with a minimum constant factor with respect to the choice of the stepsize sequence. To this end, we consider both subgradient and gradient RBSMD methods addressing nonsmooth and smooth problems, respectively. For each scheme, (i) we develop self-tuned stepsize rules characterized in terms of problem parameters and algorithm settings; (ii) we show that the non-averaging iterate generated by the underlying RBSMD method converges to the optimal solution both in an almost sure and a mean sense; (iii) we show that the mean squared error is minimized. When problem parameters are unknown, we develop a unifying self-tuned update rule that can be applied in both subgradient and gradient SMD methods, and show that for any arbitrary and small enough initial stepsize, a suitably defined error bound is minimized. We provide constant factor comparisons with standard SMD and RBSMD methods. Our numerical experiments performed on an SVM model display that the self-tuned schemes are significantly robust with respect to the choice of problem parameters, and the initial stepsize.

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arXiv2018

On stochastic mirror-prox algorithms for stochastic Cartesian variational inequalities: randomized block coordinate and optimal averaging schemes

Motivated by multi-user optimization problems and non-cooperative Nash games in uncertain regimes, we consider stochastic Cartesian variational inequalities (SCVI) where the set is given as the Cartesian product of a collection of component sets. First, we consider the case where the number of the component sets is large. For solving this type of problems, the classical stochastic approximation methods and their prox generalizations are computationally inefficient as each iteration becomes very costly. To address this challenge, we develop a randomized block stochastic mirror-prox (B-SMP) algorithm, where at each iteration only a randomly selected block coordinate of the solution is updated through implementing two consecutive projection steps. Under standard assumptions on the problem and settings of the algorithm, we show that when the mapping is strictly pseudo-monotone, the algorithm generates a sequence of iterates that converges to the solution of the problem almost surely. To derive rate statements, we assume that the maps are strongly pseudo-monotone and obtain {a non-asymptotic mean squared error $\mathcal{O}\left(\frac{d}{k}\right)$, where $k$ is the iteration number and $d$ is the number of component sets. Second, we consider large-scale stochastic optimization problems with convex objectives. For this class of problems, we develop a new averaging scheme for the B-SMP algorithm. Unlike the classical averaging stochastic mirror-prox (SMP) method where a decreasing set of weights for the averaging sequence is used, here we consider a different set of weights that are characterized in terms of the stepsizes and a {parameter}. We show that using such weights, the objective values of the averaged sequence converges to the optimal value in the mean sense with the rate $\mathcal{O}\left(\frac{\sqrt{d}}{\sqrt{k}}\right)$.

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arXiv2016

Stochastic quasi-Newton methods for non-strongly convex problems: convergence and rate analysis

Motivated by applications in optimization and machine learning, we consider stochastic quasi-Newton (SQN) methods for solving stochastic optimization problems. In the literature, the convergence analysis of these algorithms relies on strong convexity of the objective function. To our knowledge, no theoretical analysis is provided for the rate statements in the absence of this assumption. Motivated by this gap, we allow the objective function to be merely convex and we develop a cyclic regularized SQN method where the gradient mapping and the Hessian approximation matrix are both regularized at each iteration and are updated in a cyclic manner. We show that, under suitable assumptions on the stepsize and regularization parameters, the objective function value converges to the optimal objective function of the original problem in both almost sure and the expected senses. For each case, a class of feasible sequences that guarantees the convergence is provided. Moreover, the rate of convergence in terms of the objective function value is derived. Our empirical analysis on a binary classification problem shows that the proposed scheme performs well compared to both classic regularization SQN schemes and stochastic approximation method.

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arXiv2016

On Smoothing, Regularization and Averaging in Stochastic Approximation Methods for Stochastic Variational Inequalities

Traditionally, stochastic approximation schemes for SVIs have relied on strong monotonicity and Lipschitzian properties of the underlying map. In contrast, we consider monotone stochastic variational inequality (SVI) problems where the strong monotonicity and Lipschitzian assumptions on the mappings are weakened. In the first part of the paper, to address such shortcomings, a regularized smoothed SA (RSSA) scheme is developed wherein the stepsize, smoothing, and regularization parameters are diminishing sequences updated after every iteration. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to a solution in an almost sure sense, extending the results in [16] to the non-Lipschitzian regime. Motivated by the need to develop non-asymptotic rate statements, in the second part of the paper, we develop a variant of the RSSA scheme, denoted by aRSSA$_r$, in which we employ a weighted iterate-averaging, parametrized by a scalar $r$ where $r = 1$ provides us with the standard averaging scheme. We make several contributions in this context: First, we show that the gap function associated with the sequences by the aRSSA$_r$ scheme tends to zero when the parameter sequences are chosen appropriately. Second, we show that the gap function associated with the averaged sequence diminishes to zero at the optimal rate $\cal{O}(1/\sqrt{K})$ after $K$ steps when smoothing and regularization are suppressed and $r < 1$, thus improving the rate statement for the standard averaging which admits a rate of $\cal{O}(\ln(K)/\sqrt{K})$. Third, we develop a window-based variant of this scheme that also displays the optimal rate for $r < 1$. Notably, we prove the superiority of the scheme with $r < 1$ with its counterpart with $r=1$ in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large.

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arXiv2014

Optimal robust smoothing extragradient algorithms for stochastic variational inequality problems

We consider stochastic variational inequality problems where the mapping is monotone over a compact convex set. We present two robust variants of stochastic extragradient algorithms for solving such problems. Of these, the first scheme employs an iterative averaging technique where we consider a generalized choice for the weights in the averaged sequence. Our first contribution is to show that using an appropriate choice for these weights, a suitably defined gap function attains the optimal rate of convergence ${\cal O}\left(\frac{1}{\sqrt{k}}\right)$. In the second part of the paper, under an additional assumption of weak-sharpness, we update the stepsize sequence using a recursive rule that leverages problem parameters. The second contribution lies in showing that employing such a sequence, the extragradient algorithm possesses almost-sure convergence to the solution as well as convergence in a mean-squared sense to the solution of the problem at the rate ${\cal O}\left(\frac{1}{k}\right)$. Motivated by the absence of a Lipschitzian parameter, in both schemes we utilize a locally randomized smoothing scheme. Importantly, by approximating a smooth mapping, this scheme enables us to estimate the Lipschitzian parameter. The smoothing parameter is updated per iteration and we show convergence to the solution of the original problem in both algorithms.

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arXiv2013

A distributed adaptive steplength stochastic approximation method for monotone stochastic Nash Games

We consider a distributed stochastic approximation (SA) scheme for computing an equilibrium of a stochastic Nash game. Standard SA schemes employ diminishing steplength sequences that are square summable but not summable. Such requirements provide a little or no guidance for how to leverage Lipschitzian and monotonicity properties of the problem and naive choices generally do not preform uniformly well on a breadth of problems. While a centralized adaptive stepsize SA scheme is proposed in [1] for the optimization framework, such a scheme provides no freedom for the agents in choosing their own stepsizes. Thus, a direct application of centralized stepsize schemes is impractical in solving Nash games. Furthermore, extensions to game-theoretic regimes where players may independently choose steplength sequences are limited to recent work by Koshal et al. [2]. Motivated by these shortcomings, we present a distributed algorithm in which each player updates his steplength based on the previous steplength and some problem parameters. The steplength rules are derived from minimizing an upper bound of the errors associated with players' decisions. It is shown that these rules generate sequences that converge almost surely to an equilibrium of the stochastic Nash game. Importantly, variants of this rule are suggested where players independently select steplength sequences while abiding by an overall coordination requirement. Preliminary numerical results are seen to be promising.

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arXiv2013

Distributed adaptive steplength stochastic approximation schemes for Cartesian stochastic variational inequality problems

Motivated by problems arising in decentralized control problems and non-cooperative Nash games, we consider a class of strongly monotone Cartesian variational inequality (VI) problems, where the mappings either contain expectations or their evaluations are corrupted by error. Such complications are captured under the umbrella of Cartesian stochastic variational inequality problems and we consider solving such problems via stochastic approximation (SA) schemes. Specifically, we propose a scheme wherein the steplength sequence is derived by a rule that depends on problem parameters such as monotonicity and Lipschitz constants. The proposed scheme is seen to produce sequences that are guaranteed to converge almost surely to the unique solution of the problem. To cope with networked multi-agent generalizations, we provide requirements under which independently chosen steplength rules still possess desirable almost-sure convergence properties. In the second part of this paper, we consider a regime where Lipschitz constants on the map are either unavailable or difficult to derive. Here, we present a local randomization technique that allows for deriving an approximation of the original mapping, which is then shown to be Lipschitz continuous with a prescribed constant. Using this technique, we introduce a locally randomized SA algorithm and provide almost-sure convergence theory for the resulting sequence of iterates to an approximate solution of the original variational inequality problem. Finally, the paper concludes with some preliminary numerical results on a stochastic rate allocation problem and a stochastic Nash-Cournot game.

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