arXiv subjects

Jayakrishnan Nair

Publications and source records attributed to Jayakrishnan Nair.

At least 37 records · Page 2Linked to original sources

arXiv2022

Statistically Robust, Risk-Averse Best Arm Identification in Multi-Armed Bandits

Traditional multi-armed bandit (MAB) formulations usually make certain assumptions about the underlying arms' distributions, such as bounds on the support or their tail behaviour. Moreover, such parametric information is usually 'baked' into the algorithms. In this paper, we show that specialized algorithms that exploit such parametric information are prone to inconsistent learning performance when the parameter is misspecified. Our key contributions are twofold: (i) We establish fundamental performance limits of statistically robust MAB algorithms under the fixed-budget pure exploration setting, and (ii) We propose two classes of algorithms that are asymptotically near-optimal. Additionally, we consider a risk-aware criterion for best arm identification, where the objective associated with each arm is a linear combination of the mean and the conditional value at risk (CVaR). Throughout, we make a very mild 'bounded moment' assumption, which lets us work with both light-tailed and heavy-tailed distributions within a unified framework.

cs.LG↗

arXiv2021

Dynamic scheduling in a partially fluid, partially lossy queueing system

We consider a single server queueing system with two classes of jobs: eager jobs with small sizes that require service to begin almost immediately upon arrival, and tolerant jobs with larger sizes that can wait for service. While blocking probability is the relevant performance metric for the eager class, the tolerant class seeks to minimize its mean sojourn time. In this paper, we discuss the performance of each class under dynamic scheduling policies, where the scheduling of both classes depends on the instantaneous state of the system. This analysis is carried out under a certain fluid limit, where the arrival rate and service rate of the eager class are scaled to infinity, holding the offered load constant. Our performance characterizations reveal a (dynamic) pseudo-conservation law that ties the performance of both the classes to the standalone blocking probabilities of the eager class. Further, the performance is robust to other specifics of the scheduling policies. We also characterize the Pareto frontier of the achievable region of performance vectors under the same fluid limit, and identify a (two-parameter) class of Pareto-complete scheduling policies.

cs.PF↗

arXiv2021

Sequential Community Mode Estimation

We consider a population, partitioned into a set of communities, and study the problem of identifying the largest community within the population via sequential, random sampling of individuals. There are multiple sampling domains, referred to as \emph{boxes}, which also partition the population. Each box may consist of individuals of different communities, and each community may in turn be spread across multiple boxes. The learning agent can, at any time, sample (with replacement) a random individual from any chosen box; when this is done, the agent learns the community the sampled individual belongs to, and also whether or not this individual has been sampled before. The goal of the agent is to minimize the probability of mis-identifying the largest community in a \emph{fixed budget} setting, by optimizing both the sampling strategy as well as the decision rule. We propose and analyse novel algorithms for this problem, and also establish information theoretic lower bounds on the probability of error under any algorithm. In several cases of interest, the exponential decay rates of the probability of error under our algorithms are shown to be optimal up to constant factors. The proposed algorithms are further validated via simulations on real-world datasets.

stat.ML↗

arXiv2021

Coalition Formation in Constant Sum Queueing Games

We analyse a coalition formation game between strategic service providers of a congestible service. The key novelty of our formulation is that it is a constant sum game, i.e., the total payoff across all service providers (or coalitions of providers) is fixed, and dictated by the total size of the market. The game thus captures the tension between resource pooling (to benefit from the resulting statistical economies of scale) and competition between coalitions over market share. In a departure from the prior literature on resource pooling for congestible services, we show that the grand coalition is in general not stable, once we allow for competition over market share. Instead, the stable configurations are duopolies, where the dominant coalition exploits its economies of scale to corner a disproportionate market share. We analyse the stable duopolies that emerge from this interaction, and also study a dynamic variant of this game.

cs.GT↗

arXiv2021

Optimal Cycling of a Heterogenous Battery Bank via Reinforcement Learning

We consider the problem of optimal charging/discharging of a bank of heterogenous battery units, driven by stochastic electricity generation and demand processes. The batteries in the battery bank may differ with respect to their capacities, ramp constraints, losses, as well as cycling costs. The goal is to minimize the degradation costs associated with battery cycling in the long run; this is posed formally as a Markov decision process. We propose a linear function approximation based Q-learning algorithm for learning the optimal solution, using a specially designed class of kernel functions that approximate the structure of the value functions associated with the MDP. The proposed algorithm is validated via an extensive case study.

cs.LG↗

arXiv2021

Speed Scaling with Multiple Servers Under A Sum Power Constraint

The problem of scheduling jobs and choosing their respective speeds with multiple servers under a sum power constraint to minimize the flow time + energy is considered. This problem is a generalization of the flow time minimization problem with multiple unit-speed servers, when jobs can be parallelized, however, with a sub-linear, concave speedup function $k^{1/α}, α>1$ when allocated $k$ servers, i.e., jobs experience diminishing returns from being allocated additional servers. When all jobs are available at time $0$, we show that a very simple algorithm EQUI, that processes all available jobs at the same speed is $\left(2-\frac{1}α\right) \frac{2}{\left(1-\left(\frac{1}α\right)\right)}$-competitive, while in the general case, when jobs arrive over time, an LCFS based algorithm is shown to have a constant (dependent only on $α$) competitive ratio.

cs.DS↗

arXiv2021

Speed Scaling On Parallel Servers with MapReduce Type Precedence Constraints

A multiple server setting is considered, where each server has tunable speed, and increasing the speed incurs an energy cost. Jobs arrive to a single queue, and each job has two types of sub-tasks, map and reduce, and a {\bf precedence} constraint among them: any reduce task of a job can only be processed once all the map tasks of the job have been completed. In addition to the scheduling problem, i.e., which task to execute on which server, with tunable speed, an additional decision variable is the choice of speed for each server, so as to minimize a linear combination of the sum of the flow times of jobs/tasks and the total energy cost. The precedence constraints present new challenges for the speed scaling problem with multiple servers, namely that the number of tasks that can be executed at any time may be small but the total number of outstanding tasks might be quite large. We present simple speed scaling algorithms that are shown to have competitive ratios, that depend on the power cost function, and/or the ratio of the size of the largest task and the shortest reduce task, but not on the number of jobs, or the number of servers.

cs.DS↗

arXiv2020

Bandit algorithms: Letting go of logarithmic regret for statistical robustness

We study regret minimization in a stochastic multi-armed bandit setting and establish a fundamental trade-off between the regret suffered under an algorithm, and its statistical robustness. Considering broad classes of underlying arms' distributions, we show that bandit learning algorithms with logarithmic regret are always inconsistent and that consistent learning algorithms always suffer a super-logarithmic regret. This result highlights the inevitable statistical fragility of all `logarithmic regret' bandit algorithms available in the literature---for instance, if a UCB algorithm designed for $σ$-subGaussian distributions is used in a subGaussian setting with a mismatched variance parameter, the learning performance could be inconsistent. Next, we show a positive result: statistically robust and consistent learning performance is attainable if we allow the regret to be slightly worse than logarithmic. Specifically, we propose three classes of distribution oblivious algorithms that achieve an asymptotic regret that is arbitrarily close to logarithmic.

cs.LG↗

arXiv2020

Pareto-optimal energy sharing between battery-equipped renewable generators

The inherent intermittency of renewable sources like wind and solar has resulted in a bundling of renewable generators with storage resources (batteries) for increased reliability. In this paper, we consider the problem of energy sharing between two such bundles, each associated with their own demand profiles. The demand profiles might, for example, correspond to commitments made by the bundle to the grid. With each bundle seeking to minimize its loss of load rate, we explore the possibility that one bundle can supply energy to the other from its battery at times of deficit, in return for a reciprocal supply from the other when it faces a deficit itself. We show that there always exist \emph{mutually beneficial} energy sharing arrangements between the two bundles. Moreover, we show that Pareto-optimal arrangements involve at least one bundle transferring energy to the other at the maximum feasible rate at times of deficit. We illustrate the potential gains from such dynamic energy sharing via an extensive case study.

eess.SY↗

arXiv2020

Multiple Server SRPT with speed scaling is competitive

Can the popular shortest remaining processing time (SRPT) algorithm achieve a constant competitive ratio on multiple servers when server speeds are adjustable (speed scaling) with respect to the flow time plus energy consumption metric? This question has remained open for a while, where a negative result in the absence of speed scaling is well known. The main result of this paper is to show that multi-server SRPT can be constant competitive, with a competitive ratio that only depends on the power-usage function of the servers, but not on the number of jobs/servers or the job sizes (unlike when speed scaling is not allowed). When all job sizes are unity, we show that round-robin routing is optimal and can achieve the same competitive ratio as the best known algorithm for the single server problem. Finally, we show that a class of greedy dispatch policies, including policies that route to the least loaded or the shortest queue, do not admit a constant competitive ratio. When job arrivals are stochastic, with Poisson arrivals and i.i.d. job sizes, we show that random routing and a simple gated-static speed scaling algorithm achieves a constant competitive ratio.

cs.PF↗

arXiv2019

Adaptive flow-level scheduling for the IoT MAC

Over the past decade, distributed CSMA, which forms the basis for WiFi, has been deployed ubiquitously to provide seamless and high-speed mobile internet access. However, distributed CSMA might not be ideal for future IoT/M2M applications, where the density of connected devices/sensors/controllers is expected to be orders of magnitude higher than that in present wireless networks. In such high-density networks, the overhead associated with completely distributed MAC protocols will become a bottleneck. Moreover, IoT communications are likely to have strict QoS requirements, for which the `best-effort' scheduling by present WiFi networks may be unsuitable. This calls for a clean-slate redesign of the wireless MAC taking into account the requirements for future IoT/M2M networks. In this paper, we propose a reservation-based (for minimal overhead) wireless MAC designed specifically with IoT/M2M applications in mind. The key features include: (i) flow-level, rather than packet level contention to minimize overhead, (ii) deadline aware, reservation based scheduling, and (iii) the ability to dynamically adapt the MAC parameters with changing workload.

cs.NI↗

arXiv2019

Statistical Economies of Scale in Battery Sharing

The goal of this paper is to shed light on the statistical economies of scale achievable from sharing of storage between renewable generators. We conduct an extensive study using real world wind data from a grid of equispaced wind generators sharing a common battery. We assume each generator is contracted to meet a certain demand profile to a prescribed level of reliability. We find that the statistical diversity in wind generation across different locations yields useful economies of scale once the grid spacing exceeds 200 km. When the grid spacing exceeds 500 km, we find that the economies grow dramatically: The shared battery size becomes insensitive to the number of participating generators. This means that the generators can access a common, shared battery and collectively achieve the same reliability they would have, had each of them had the entire battery to themselves. To provide a rigourous foundation for this remarkable observation, we propose a mathematical model that demonstrates this phenomenon, assuming that the net generation (generation minus demand) processes associated with the generators are statistically independent. The result is derived by characterizing the large deviations exponent of the loss of load probability with increasing battery size, and showing that this exponent is invariant with the number of generators.

eess.SY↗

arXiv2019

Please come back later: Benefiting from deferrals in service systems

The performance evaluation of loss service systems, where customers who cannot be served upon arrival get dropped, has a long history going back to the classical Erlang B model. In this paper, we consider the performance benefits arising from the possibility of deferring customers who cannot be served upon arrival. Specifically, we consider an Erlang B type loss system where the system operator can, subject to certain constraints, ask a customer arriving when all servers are busy, to come back at a specified time in the future. If the system is still fully loaded when the deferred customer returns, she gets dropped for good. For such a system, we ask: How should the system operator determine the rearrival times of the deferred customers based on the state of the system (which includes those customers already deferred and yet to arrive)? How does one quantify the performance benefit of such a deferral policy? Our contributions are as follows. We propose a simple state-dependent policy for determining the rearrival times of deferred customers. For this policy, we characterize the long run fraction of customers dropped. We also analyse a relaxation where the deferral times are bounded in expectation. Via extensive numerical evaluations, we demonstrate the superiority of the proposed state-dependent policies over naive state-independent deferral policies.

cs.PF↗

arXiv2019

Revenue Sharing in the Internet: A Moral Hazard Approach and a Net-neutrality Perspective

Revenue sharing contracts between Content Providers (CPs) and Internet Service Providers (ISPs) can act as leverage for enhancing the infrastructure of the Internet. ISPs can be incentivized to make investments in network infrastructure that improve Quality of Service (QoS) for users if attractive contracts are negotiated between them and CPs. The idea here is that part of the net profit gained by CPs are given to ISPs to invest in the network. The Moral Hazard economic framework is used to model such an interaction, in which a principal determines a contract, and an agent reacts by adapting her effort. In our setting, several competitive CPs interact through one common ISP. Two cases are studied: (i) the ISP differentiates between the CPs and makes a (potentially) different investment to improve the QoS of each CP, and (ii) the ISP does not differentiate between CPs and makes a common investment for both. The last scenario can be viewed as \emph{network neutral behavior} on the part of the ISP. We analyse the optimal contracts and show that the CP that can better monetize its demand always prefers the non-neutral regime. Interestingly, ISP revenue, as well as social utility, are also found to be higher under the non-neutral regime.

econ.GN↗

arXiv2019

Efficiency Fairness Tradeoff in Battery Sharing

The increasing presence of decentralized renewable generation in the power grid has motivated consumers to install batteries to save excess energy for future use. The high price of energy storage calls for a shared storage system, but careful battery management is required so that the battery is operated in a manner that is fair to all and as efficiently as possible. In this paper, we study the tradeoffs between efficiency and fairness in operating a shared battery. We develop a framework based on constrained Markov decision processes to study both regimes, namely, optimizing efficiency under a hard fairness constraint and optimizing fairness under hard efficiency constraint. Our results show that there are fundamental limits to efficiency under fairness and vice-versa, and, in general, the two cannot be achieved simultaneously. We characterize these fundamental limits via absolute bounds on these quantities, and via the notion of price of fairness that we introduce in this paper.

math.OC↗

arXiv2019

Speed Scaling with Tandem Servers

Speed scaling for a tandem server setting is considered, where there is a series of servers, and each job has to be processed by each of the servers in sequence. Servers have a variable speed, their power consumption being a convex increasing function of the speed. We consider the worst case setting as well as the stochastic setting. In the worst case setting, the jobs are assumed to be of unit size with arbitrary (possibly adversarially determined) arrival instants. For this problem, we devise an online speed scaling algorithm that is constant competitive with respect to the optimal offline algorithm that has non-causal information. The proposed algorithm, at all times, uses the same speed on all active servers, such that the total power consumption equals the number of outstanding jobs. In the stochastic setting, we consider a more general tandem network, with a parallel bank of servers at each stage. In this setting, we show that random routing with a simple gated static speed selection is constant competitive. In both cases, the competitive ratio depends only on the power functions, and is independent of the workload and the number of servers.

cs.DS↗

arXiv2019

Partial Server Pooling in Redundancy Systems

Partial sharing allows providers to possibly pool a fraction of their resources when full pooling is not beneficial to them. Recent work in systems without sharing has shown that redundancy can improve performance considerably. In this paper, we combine partial sharing and redundancy by developing partial sharing models for providers operating multi-server systems with redundancy. Two M/M/N queues with redundant service models are considered. Copies of an arriving job are placed in the queues of servers that can serve the job. Partial sharing models for cancel-on-complete and cancel-on-start redundancy models are developed. For cancel-on-complete, it is shown that the Pareto efficient region is the full pooling configuration. For a cancel-on-start policy, we conjecture that the Pareto frontier is always non-empty and is such that at least one of the two providers is sharing all of its resources. For this system, using bargaining theory the sharing configuration that the providers may use is determined. Mean response time and probability of waiting are the performance metrics considered.

cs.NI↗

arXiv2019

Distribution oblivious, risk-aware algorithms for multi-armed bandits with unbounded rewards

Classical multi-armed bandit problems use the expected value of an arm as a metric to evaluate its goodness. However, the expected value is a risk-neutral metric. In many applications like finance, one is interested in balancing the expected return of an arm (or portfolio) with the risk associated with that return. In this paper, we consider the problem of selecting the arm that optimizes a linear combination of the expected reward and the associated Conditional Value at Risk (CVaR) in a fixed budget best-arm identification framework. We allow the reward distributions to be unbounded or even heavy-tailed. For this problem, our goal is to devise algorithms that are entirely distribution oblivious, i.e., the algorithm is not aware of any information on the reward distributions, including bounds on the moments/tails, or the suboptimality gaps across arms. In this paper, we provide a class of such algorithms with provable upper bounds on the probability of incorrect identification. In the process, we develop a novel estimator for the CVaR of unbounded (including heavy-tailed) random variables and prove a concentration inequality for the same, which could be of independent interest. We also compare the error bounds for our distribution oblivious algorithms with those corresponding to standard non-oblivious algorithms. Finally, numerical experiments reveal that our algorithms perform competitively when compared with non-oblivious algorithms, suggesting that distribution obliviousness can be realised in practice without incurring a significant loss of performance.

cs.LG↗