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Ari Arapostathis

Publications and source records attributed to Ari Arapostathis.

50 records · Page 3Linked to original sources
arXiv2015

On Solutions of Mean Field Games with Ergodic Cost

A general class of mean field games are considered where the governing dynamics are controlled diffusions in $\mathbb{R}^d$. The optimization criterion is the long time average of a running cost function. Under various sets of hypotheses, we establish the existence of mean field game solutions. We also study the long time behavior of mean field game solutions associated with the finite horizon problem, and under the assumption of geometric ergodicity for the dynamics, we show that these converge to the ergodic mean field game solution as the horizon tends to infinity. Lastly, we study the associated $N$-player games, show existence of Nash equilibria, and establish the convergence of the solutions associated to Nash equilibria of the game to a mean field game solution as $N\to\infty$.

math.OC↗
arXiv2015

Ergodic control of multi-class $M/M/N+M$ queues in the Halfin-Whitt regime

We study a dynamic scheduling problem for a multi-class queueing network with a large pool of statistically identical servers. The arrival processes are Poisson, and service times and patience times are assumed to be exponentially distributed and class dependent. The optimization criterion is the expected long time average (ergodic) of a general (nonlinear) running cost function of the queue lengths. We consider this control problem in the Halfin-Whitt (QED) regime, that is, the number of servers $n$ and the total offered load $\mathbf{r}$ scale like $n\approx\mathbf{r}+\hatρ\sqrt{\mathbf{r}}$ for some constant $\hatρ$. This problem was proposed in [Ann. Appl. Probab. 14 (2004) 1084-1134, Section 5.2]. The optimal solution of this control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. We introduce a broad class of ergodic control problems for controlled diffusions, which includes a large class of queueing models in the diffusion approximation, and establish a complete characterization of optimality via the study of the associated HJB equation. We also prove the asymptotic convergence of the values for the multi-class queueing control problem to the value of the associated ergodic diffusion control problem. The proof relies on an approximation method by spatial truncation for the ergodic control of diffusion processes, where the Markov policies follow a fixed priority policy outside a fixed compact set.

math.PR↗
arXiv2015

On a Class of Stochastic Differential Equations With Jumps and Its Properties

We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for a class of these operators. We also establish key connections between the recurrence properties of these jump processes and the non-local partial differential operator. One of the key results is the regularity of solutions of the Dirichlet problem for a class of operators with locally weakly Hölder continuous kernels.

math.PR↗
arXiv2015

Ergodic Diffusion Control of Multiclass Multi-Pool Networks in the Halfin-Whitt Regime

We consider Markovian multiclass multi-pool networks with heterogeneous server pools, each consisting of many statistically identical parallel servers, where the bipartite graph of customer classes and server pools forms a tree. Customers form their own queue and are served in the first-come first-served discipline, and can abandon while waiting in queue. Service rates are both class and pool dependent. The objective is to study the limiting diffusion control problems under the long run average (ergodic) cost criteria in the Halfin--Whitt regime. Two formulations of ergodic diffusion control problems are considered: (i) both queueing and idleness costs are minimized, and (ii) only the queueing cost is minimized while a constraint is imposed upon the idleness of all server pools. We develop a recursive leaf elimination algorithm that enables us to obtain an explicit representation of the drift for the controlled diffusions. Consequently, we show that for the limiting controlled diffusions, there always exists a stationary Markov control under which the diffusion process is geometrically ergodic. The framework developed in our earlier work is extended to address a broad class of ergodic diffusion control problems with constraints. We show that that the unconstrained and constrained problems are well posed, and we characterize the optimal stationary Markov controls via HJB equations.

math.PR↗
arXiv2015

Risk-Sensitive Control and an Abstract Collatz-Wielandt Formula

The "value" of infinite horizon risk-sensitive control is the principal eigenvalue of a certain positive operator. This facilitates the use of Chang's extension of the Collatz-Wielandt formula to derive a variational characterization thereof. For the uncontrolled case, this reduces to the Donsker-Varadhan functional.

math.OC↗
arXiv2013

Relative Value Iteration for Stochastic Differential Games

We study zero-sum stochastic differential games with player dynamics governed by a nondegenerate controlled diffusion process. Under the assumption of uniform stability, we establish the existence of a solution to the Isaac's equation for the ergodic game and characterize the optimal stationary strategies. The data is not assumed to be bounded, nor do we assume geometric ergodicity. Thus our results extend previous work in the literature. We also study a relative value iteration scheme that takes the form of a parabolic Isaac's equation. Under the hypothesis of geometric ergodicity we show that the relative value iteration converges to the elliptic Isaac's equation as time goes to infinity. We use these results to establish convergence of the relative value iteration for risk-sensitive control problems under an asymptotic flatness assumption.

math.OC↗
arXiv2013

Convergence of The Relative Value Iteration for the Ergodic Control Problem of Nondegenerate Diffusions under Near-Monotone Costs

We study the relative value iteration for the ergodic control problem under a near-monotone running cost structure for a nondegenerate diffusion controlled through its drift. This algorithm takes the form of a quasilinear parabolic Cauchy initial value problem in $\RR^{d}$. We show that this Cauchy problem stabilizes, or in other words, that the solution of the quasilinear parabolic equation converges for every bounded initial condition in $\Cc^{2}(\RR^{d})$ to the solution of the Hamilton--Jacobi--Bellman (HJB) equation associated with the ergodic control problem.

math.OC↗
arXiv2012

Resource Allocation: Realizing Mean-Variability-Fairness Tradeoffs

Network Utility Maximization (NUM) provides a key conceptual framework to study reward allocation amongst a collection of users/entities across disciplines as diverse as economics, law and engineering. In network engineering, this framework has been particularly insightful towards understanding how Internet protocols allocate bandwidth, and motivated diverse research efforts on distributed mechanisms to maximize network utility while incorporating new relevant constraints, on energy, power, storage, stability, etc., e.g., for systems ranging from communication networks to the smart-grid. However when the available resources and/or users' utilities vary over time, reward allocations will tend to vary, which in turn may have a detrimental impact on the users' overall satisfaction or quality of experience. This paper introduces a generalization of NUM framework which explicitly incorporates the detrimental impact of temporal variability in a user's allocated rewards. It explicitly incorporates tradeoffs amongst the mean and variability in users' reward allocations, as well as fairness. We propose a simple online algorithm to realize these tradeoffs, which, under stationary ergodic assumptions, is shown to be asymptotically optimal, i.e., achieves a long term performance equal to that of an offline algorithm with knowledge of the future variability in the system. This substantially extends work on NUM to an interesting class of relevant problems where users/entities are sensitive to temporal variability in their service or allocated rewards.

eess.SY↗
arXiv2011

Aspiration Learning in Coordination Games

We consider the problem of distributed convergence to efficient outcomes in coordination games through dynamics based on aspiration learning. Under aspiration learning, a player continues to play an action as long as the rewards received exceed a specified aspiration level. Here, the aspiration level is a fading memory average of past rewards, and these levels also are subject to occasional random perturbations. A player becomes dissatisfied whenever a received reward is less than the aspiration level, in which case the player experiments with a probability proportional to the degree of dissatisfaction. Our first contribution is the characterization of the asymptotic behavior of the induced Markov chain of the iterated process in terms of an equivalent finite-state Markov chain. We then characterize explicitly the behavior of the proposed aspiration learning in a generalized version of coordination games, examples of which include network formation and common-pool games. In particular, we show that in generic coordination games the frequency at which an efficient action profile is played can be made arbitrarily large. Although convergence to efficient outcomes is desirable, in several coordination games, such as common-pool games, attainability of fair outcomes, i.e., sequences of plays at which players experience highly rewarding returns with the same frequency, might also be of special interest. To this end, we demonstrate through analysis and simulations that aspiration learning also establishes fair outcomes in all symmetric coordination games, including common-pool games.

cs.GT↗
arXiv2011

A Relative Value Iteration Algorithm for Non-degenerate Controlled Diffusions

The ergodic control problem for a non-degenerate controlled diffusion controlled through its drift is considered under a uniform stability condition that ensures the well-posedness of the associated Hamilton-Jacobi-Bellman (HJB) equation. A nonlinear parabolic evolution equation is then proposed as a continuous time continuous state space analog of White's `relative value iteration' algorithm for solving the ergodic dynamic programming equation for the finite state finite action case. Its convergence to the solution of the HJB equation is established using the theory of monotone dynamical systems and also, alternatively, by using the theory of reverse martingales.

math.OC↗
arXiv2006

On the Capacity of Multiple Access Channels with State Information and Feedback

In this paper, the multiple access channel (MAC) with channel state is analyzed in a scenario where a) the channel state is known non-causally to the transmitters and b) there is perfect causal feedback from the receiver to the transmitters. An achievable region and an outer bound are found for a discrete memoryless MAC that extend existing results, bringing together ideas from the two separate domains of MAC with state and MAC with feedback. Although this achievable region does not match the outer bound in general, special cases where they meet are identified. In the case of a Gaussian MAC, a specialized achievable region is found by using a combination of dirty paper coding and a generalization of the Schalkwijk-Kailath, Ozarow and Merhav-Weissman schemes, and this region is found to be capacity achieving. Specifically, it is shown that additive Gaussian interference that is known non-causally to the transmitter causes no loss in capacity for the Gaussian MAC with feedback.

cs.IT↗
arXiv2006

On the Capacity of Interference Channels with Degraded Message sets

This paper is motivated by a sensor network on a correlated field where nearby sensors share information, and can thus assist rather than interfere with one another. A special class of two-user Gaussian interference channels (IFCs) is considered where one of the two transmitters knows both the messages to be conveyed to the two receivers (called the IFC with degraded message sets). Both achievability and converse arguments are provided for this scenario for a class of discrete memoryless channels with weak interference. For the case of the Gaussian weak interference channel with degraded message sets, optimality of Gaussian inputs is also shown, resulting in the capacity region of this channel.

cs.IT↗
arXiv2006

On the Capacity of Gaussian Weak Interference Channels with Degraded Message sets

This paper is motivated by a sensor network on a correlated field where nearby sensors share information, and can thus assist rather than interfere with one another. We consider a special class of two-user Gaussian interference channels (IFCs) where one of the two transmitters knows both the messages to be conveyed to the two receivers. Both achievability and converse arguments are provided for a channel with Gaussian inputs and Gaussian noise when the interference is weaker than the direct link (a so called weak IFC). In general, this region serves as an outer bound on the capacity of weak IFCs with no shared knowledge between transmitters.

cs.IT↗