arXiv subjects

Uday V. Shanbhag

Publications and source records attributed to Uday V. Shanbhag.

48 records · Page 3Linked to original sources

arXiv2016

Distributed Algorithms for Aggregative Games on Graphs

We consider a class of Nash games, termed as aggregative games, being played over a networked system. In an aggregative game, a player's objective is a function of the aggregate of all the players' decisions. Every player maintains an estimate of this aggregate, and the players exchange this information with their local neighbors over a connected network. We study distributed synchronous and asynchronous algorithms for information exchange and equilibrium computation over such a network. Under standard conditions, we establish the almost-sure convergence of the obtained sequences to the equilibrium point. We also consider extensions of our schemes to aggregative games where the players' objectives are coupled through a more general form of aggregate function. Finally, we present numerical results that demonstrate the performance of the proposed schemes.

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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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arXiv2015

On robust solutions to uncertain linear complementarity problems and their variants

A popular approach for addressing uncertainty in variational inequality problems is by solving the expected residual minimization (ERM) problem. This avenue necessitates distributional information associated with the uncertainty and requires minimizing nonconvex expectation-valued functions. We consider a distinctly different approach in the context of uncertain linear complementarity problems with a view towards obtaining robust solutions. Specifically, we define a robust solution to a complementarity problem as one that minimizes the worst-case of the gap function. In what we believe is amongst the first efforts to comprehensively address such problems in a distribution-free environment, we show that under specified assumptions on the uncertainty sets, the robust solutions to uncertain monotone linear complementarity problem can be tractably obtained through the solution of a single convex program. We also define uncertainty sets that ensure that robust solutions to non-monotone generalizations can also be obtained by solving convex programs. More generally, robust counterparts of uncertain non-monotone LCPs are proven to be low-dimensional nonconvex quadratically constrained quadratic programs. We show that these problems may be globally resolved by customizing an existing branching scheme. We further extend the tractability results to include uncertain affine variational inequality problems defined over uncertain polyhedral sets as well as to hierarchical regimes captured by mathematical programs with uncertain complementarity constraints. Preliminary numerics on uncertain linear complementarity and traffic equilibrium problems suggest that the presented avenues hold promise.

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arXiv2015

Distributed Stochastic Optimization under Imperfect Information

We consider a stochastic convex optimization problem that requires minimizing a sum of misspecified agentspecific expectation-valued convex functions over the intersection of a collection of agent-specific convex sets. This misspecification is manifested in a parametric sense and may be resolved through solving a distinct stochastic convex learning problem. Our interest lies in the development of distributed algorithms in which every agent makes decisions based on the knowledge of its objective and feasibility set while learning the decisions of other agents by communicating with its local neighbors over a time-varying connectivity graph. While a significant body of research currently exists in the context of such problems, we believe that the misspecified generalization of this problem is both important and has seen little study, if at all. Accordingly, our focus lies on the simultaneous resolution of both problems through a joint set of schemes that combine three distinct steps: (i) An alignment step in which every agent updates its current belief by averaging over the beliefs of its neighbors; (ii) A projected (stochastic) gradient step in which every agent further updates this averaged estimate; and (iii) A learning step in which agents update their belief of the misspecified parameter by utilizing a stochastic gradient step. Under an assumption of mere convexity on agent objectives and strong convexity of the learning problems, we show that the sequences generated by this collection of update rules converge almost surely to the solution of the correctly specified stochastic convex optimization problem and the stochastic learning problem, respectively.

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arXiv2015

On the solution of stochastic optimization and variational problems in imperfect information regimes

We consider the solution of a stochastic convex optimization problem $\mathbb{E}[f(x;θ^*,ξ)]$ over a closed and convex set $X$ in a regime where $θ^*$ is unavailable and $ξ$ is a suitably defined random variable. Instead, $θ^*$ may be obtained through the solution of a learning problem that requires minimizing a metric $\mathbb{E}[g(θ;η)]$ in $θ$ over a closed and convex set $Θ$. Traditional approaches have been either sequential or direct variational approaches. In the case of the former, this entails the following steps: (i) a solution to the learning problem, namely $θ^*$, is obtained; and (ii) a solution is obtained to the associated computational problem which is parametrized by $θ^*$. Such avenues prove difficult to adopt particularly since the learning process has to be terminated finitely and consequently, in large-scale instances, sequential approaches may often be corrupted by error. On the other hand, a variational approach requires that the problem may be recast as a possibly non-monotone stochastic variational inequality problem in the $(x,θ)$ space; but there are no known first-order stochastic approximation schemes are currently available for the solution of this problem. To resolve the absence of convergent efficient schemes, we present a coupled stochastic approximation scheme which simultaneously solves both the computational and the learning problems. The obtained schemes are shown to be equipped with almost sure convergence properties in regimes when the function $f$ is either strongly convex as well as merely convex.

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arXiv2015

On the resolution of misspecified convex optimization and monotone variational inequality problems

We consider a misspecified optimization problem that requires minimizing a function f(x;q*) over a closed and convex set X where q* is an unknown vector of parameters that may be learnt by a parallel learning process. In this context, We examine the development of coupled schemes that generate iterates {x_k,q_k} as k goes to infinity, then {x_k} converges x*, a minimizer of f(x;q*) over X and {q_k} converges to q*. In the first part of the paper, we consider the solution of problems where f is either smooth or nonsmooth under various convexity assumptions on function f. In addition, rate statements are also provided to quantify the degradation in rate resulted from learning process. In the second part of the paper, we consider the solution of misspecified monotone variational inequality problems to contend with more general equilibrium problems as well as the possibility of misspecification in the constraints. We first present a constant steplength misspecified extragradient scheme and prove its asymptotic convergence. This scheme is reliant on problem parameters (such as Lipschitz constants)and leads us to present a misspecified variant of iterative Tikhonov regularization. Numerics support the asymptotic and rate statements.

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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 Shared-Constraint Approach to Multi-leader Multi-follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria for it has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over the shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.

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arXiv2013

An Existence Result for Hierarchical Stackelberg v/s Stackelberg Games

In Stackelberg v/s Stackelberg games a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problems are plagued by the nonuniqueness of follower equilibria and nonconvexity of leader problems whereby the problem of providing sufficient conditions for existence of global or even local equilibria remains largely open. Indeed available existence statements are restrictive and model specific. In this paper, we present what is possibly the first general existence result for equilibria for this class of games. Importantly, we impose no single-valuedness assumption on the equilibrium of the follower-level game. Specifically, under the assumption that the objectives of the leaders admit a quasi-potential function, a concept we introduce in this paper, the global and local minimizers of a suitably defined optimization problem are shown to be the global and local equilibria of the game. In effect existence of equilibria can be guaranteed by the solvability of an optimization problem, which holds under mild and verifiable conditions. We motivate quasi- potential games through an application in communication networks.

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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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arXiv2011

On Stochastic Gradient and Subgradient Methods with Adaptive Steplength Sequences

The performance of standard stochastic approximation implementations can vary significantly based on the choice of the steplength sequence, and in general, little guidance is provided about good choices. Motivated by this gap, in the first part of the paper, we present two adaptive steplength schemes for strongly convex differentiable stochastic optimization problems, equipped with convergence theory. The first scheme, referred to as a recursive steplength stochastic approximation scheme, optimizes the error bounds to derive a rule that expresses the steplength at a given iteration as a simple function of the steplength at the previous iteration and certain problem parameters. This rule is seen to lead to the optimal steplength sequence over a prescribed set of choices. The second scheme, termed as a cascading steplength stochastic approximation scheme, maintains the steplength sequence as a piecewise-constant decreasing function with the reduction in the steplength occurring when a suitable error threshold is met. In the second part of the paper, we allow for nondifferentiable objective and we propose a local smoothing technique that leads to a differentiable approximation of the function. Assuming a uniform distribution on the local randomness, we establish a Lipschitzian property for the gradient of the approximation and prove that the obtained Lipschitz bound grows at a modest rate with problem size. This facilitates the development of an adaptive steplength stochastic approximation framework, which now requires sampling in the product space of the original measure and the artificially introduced distribution. The resulting adaptive steplength schemes are applied to three stochastic optimization problems. We observe that both schemes perform well in practice and display markedly less reliance on user-defined parameters.

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