arXiv subjects

Martin Zubeldia

Publications and source records attributed to Martin Zubeldia.

15 recordsLinked to original sources

arXiv2026

Load Balancing Policies in Heterogeneous Systems: Non-Monotone Stability and Heavy-Traffic Optimality

We consider a discrete-time queueing system with $n$ heterogeneous parallel single-server queues. Jobs arrive at a central dispatcher and must be assigned immediately to one of the queues. We develop a unified framework for a broad family of load-balancing policies, including Join the Shortest Queue (JSQ), Join the Shortest Expected Delay (JSED), and Power-of-$d$ Choices (Po$d$). In this framework, the dispatcher updates queue-length information periodically, possibly at arbitrarily long intervals, and dispatches jobs based on the sampled permutation of scaled queue lengths and the servers' service rates. Leveraging this structure, we derive a closed-form, easily verifiable sufficient condition for stability. We further show that, for general policies, stability above the induced threshold need not be monotone in the arrival rate, and we obtain an exact characterization under a persistent bottleneck dominance condition. When the stability condition holds strictly, we prove state-space collapse and heavy-traffic delay optimality. We also show that the steady-state queue-length vector converges in distribution to a deterministic vector scaled by an exponential random variable in heavy traffic. Methodologically, we extend Lyapunov-drift and transform techniques to a cycle-based analysis with multi-step updates. Our results connect the policy-induced dispatch fractions and sampled permutations to stability, delay, and distributional performance, providing guidance for designing scalable load-balancing schemes with limited queue-length information.

cs.PF↗

arXiv2026

Concentration of General Stochastic Approximation Under Heavy-Tailed Markovian Noise

We establish maximal concentration bounds for the iterates generated by stochastic approximation algorithms with general step sizes, where the noise has a finite-state Markovian component plus a Martingale-difference component. When the Martingale-difference noise is bounded, we show that the tail of the error can be sub-Gaussian, sub-Weibull, or something lighter than any Pareto but heavier than any Weibull, depending on the step size sequence and on whether the random operator is almost surely contractive, almost surely non-expansive, or expansive with positive probability. Our analysis relies on a novel Lyapunov function involving the moment-generating function of the solution to a Poisson equation, together with an auxiliary projected algorithm. We complement the upper bounds with worst-case examples showing that qualitatively sharper bounds are impossible. We further study the case of unbounded Martingale-difference noise when the average operator is contractive, and the step sizes are of order $1/k$. In this setting, we show that if the random operator is almost surely non-expansive, then the error tail is at most three times heavier than the noise tail, whereas if the random operator is expansive with positive probability, then the error may have substantially heavier tails. These results are obtained through a novel black-box truncation argument that reduces the unbounded-noise setting to the bounded-noise case.

math.PR↗

arXiv2025

Heavy-traffic Optimality of Skip-the-Longest-Queues in Heterogeneous Service Systems

We consider a discrete-time parallel service system consisting of $n$ heterogeneous single server queues with infinite capacity. Jobs arrive to the system as an i.i.d. process with rate proportional to $n$, and must be immediately dispatched in the time slot that they arrive. The dispatcher is assumed to be able to exchange messages with the servers to obtain their queue lengths and make dispatching decisions, introducing an undesirable communication overhead. In this setting, we propose a ultra-low communication overhead load balancing policy dubbed $k$-Skip-the-$d$-Longest-Queues ($k$-SLQ-$d$), where queue lengths are only observed every $k(n-d)$ time slots and, between observations, incoming jobs are sent to a queue that is not one of the $d$ longest ones at the time that the queues were last observed. For this policy, we establish conditions on $d$ for it to be throughput optimal and we show that, under that condition, it is asymptotically delay-optimal in heavy-traffic for arbitrarily low communication overheads (i.e., for arbitrarily large $k$).

cs.PF↗

arXiv2025

A General-Purpose Theorem for High-Probability Bounds of Stochastic Approximation with Polyak Averaging

Polyak-Ruppert averaging is a widely used technique to achieve the optimal asymptotic variance of stochastic approximation (SA) algorithms, yet its high-probability performance guarantees remain underexplored in general settings. In this paper, we present a general framework for establishing non-asymptotic concentration bounds for the error of averaged SA iterates. Our approach assumes access to individual concentration bounds for the unaveraged iterates and yields a sharp bound on the averaged iterates. We also construct an example, showing the tightness of our result up to constant multiplicative factors. As direct applications, we derive tight concentration bounds for contractive SA algorithms and for algorithms such as temporal difference learning and Q-learning with averaging, obtaining new bounds in settings where traditional analysis is challenging.

stat.ML↗

arXiv2024

Concentration of Contractive Stochastic Approximation: Additive and Multiplicative Noise

In this paper, we establish maximal concentration bounds for the iterates generated by a stochastic approximation (SA) algorithm under a contractive operator with respect to some arbitrary norm (for example, the $\ell_\infty$-norm). We consider two settings where the iterates are potentially unbounded: SA with bounded multiplicative noise and SA with sub-Gaussian additive noise. Our maximal concentration inequalities state that the convergence error has a sub-Gaussian tail in the additive noise setting and a Weibull tail (which is faster than polynomial decay but could be slower than exponential decay) in the multiplicative noise setting. In addition, we provide an impossibility result showing that it is generally impossible to have sub-exponential tails under multiplicative noise. To establish the maximal concentration bounds, we develop a novel bootstrapping argument that involves bounding the moment-generating function of a modified version of the generalized Moreau envelope of the convergence error and constructing an exponential supermartingale to enable using Ville's maximal inequality. We demonstrate the applicability of our theoretical results in the context of linear SA and reinforcement learning.

cs.LG↗

arXiv2023

Matching Queues with Abandonments in Quantum Switches: Stability and Throughput Analysis

Inspired by quantum switches, we consider a discrete-time multi-way matching system with two classes of arrivals: requests for entangled pair of qubits between two nodes, and qubits from each node that can be used to serve the requests. An important feature of this model is that qubits decohere and so abandon over time. In contrast to classical server-based queueing models, the combination of queueing, server-less multi-way matching, and (potentially correlated) abandonments make the analysis a challenging problem. The primary focus of this paper is to study a simple system consisting of two types of requests and three types of qubits operating under a Max-Weight policy. In this setting, we characterize the stability region under the Max-Weight policy by adopting a two-time scale fluid limit to get a handle on the abandonments. In particular, we show that Max-Weight is throughput optimal and that it can achieve throughputs larger than the ones that can be achieved by non-idling policies when the requests are infinitely backlogged. Moreover, despite the use of the Max-Weight policy, we show that there can be a counter-intuitive behavior in the system: the longest requests queue can have a positive drift for some time even if the overall system is stable.

cs.PF↗

arXiv2023

Distributed Rate Scaling in Large-Scale Service Systems

We consider a large-scale parallel-server system, where each server independently adjusts its processing speed in a decentralized manner. The objective is to minimize the overall cost, which comprises the average cost of maintaining the servers' processing speeds and a non-decreasing function of the tasks' sojourn times. The problem is compounded by the lack of knowledge of the task arrival rate and the absence of a centralized control or communication among the servers. We draw on ideas from stochastic approximation and present a novel rate scaling algorithm that ensures convergence of all server processing speeds to the globally asymptotically optimum value as the system size increases. Apart from the algorithm design, a key contribution of our approach lies in demonstrating how concepts from the stochastic approximation literature can be leveraged to effectively tackle learning problems in large-scale, distributed systems. En route, we also analyze the performance of a fully heterogeneous parallel-server system, where each server has a distinct processing speed, which might be of independent interest.

math.OC↗

arXiv2023

Join-the-Shortest Queue with Abandonment: Critically Loaded and Heavily Overloaded Regimes

We consider a load balancing system comprised of a fixed number of single server queues, operating under the well-known Join-the-Shortest Queue policy, and where jobs/customers are impatient and abandon if they do not receive service after some (random) amount of time. In this setting, we characterize the centered and appropriately scaled steady state queue length distribution (hereafter referred to as limiting distribution), in the limit as the abandonment rate goes to zero at the same time as the load either converges to one or is larger than one. Depending on the arrival, service, and abandonment rates, we observe three different regimes of operation that yield three different limiting distributions. The first regime is when the system is underloaded and its load converges relatively slowly to one. In this case, abandonments do not affect the limiting distribution, and we obtain the same exponential distribution as in the system without abandonments. When the load converges to one faster, we have the second regime, where abandonments become significant. Here, the system undergoes a phase transition, and the limiting distribution is a truncated Gaussian. Further, the third regime is when the system is heavily overloaded, and so the queue lengths are very large. In this case, we show the limiting distribution converges to a normal distribution. To establish our results, we first prove a weaker form of State Space Collapse by providing a uniform bound on the second moment of the (unscaled) perpendicular component of the queue lengths, which shows that the system behaves like a single server queue. We then use exponential Lyapunov functions to characterize the limiting distribution of the steady state queue length vector.

math.PR↗

arXiv2020

Large deviations for acyclic networks of queues with correlated Gaussian inputs

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the "overflow probability". In particular, we first leverage Schilder's sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are non-negatively correlated, non short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.

math.PR↗

arXiv2020

Stability, memory, and messaging tradeoffs in heterogeneous service systems

We consider a heterogeneous distributed service system, consisting of $n$ servers with unknown and possibly different processing rates. Jobs with unit mean and independent processing times arrive as a renewal process of rate $λn$, with $0<λ<1$, to the system. Incoming jobs are immediately dispatched to one of several queues associated with the $n$ servers. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate) in order for a dispatching policy to be {\bf maximally stable}, i.e., stable whenever the processing rates are such that the arrival rate is less than the total available processing rate. First, for the case of Poisson arrivals and exponential service times, we present a policy that is maximally stable while using a positive (but arbitrarily small) message rate, and $\log_2(n)$ bits of memory. Second, we show that within a certain broad class of policies, a dispatching policy that exchanges $o\big(n^2\big)$ messages per unit of time, and with $o(\log(n))$ bits of memory, cannot be maximally stable. Thus, as long as the message rate is not too excessive, a logarithmic memory is necessary and sufficient for maximal stability.

cs.PF↗

arXiv2019

Delay-optimal policies in partial fork-join systems with redundancy and random slowdowns

We consider a large distributed service system consisting of $n$ homogeneous servers with infinite capacity FIFO queues. Jobs arrive as a Poisson process of rate $λn/k_n$ (for some positive constant $λ$ and integer $k_n$). Each incoming job consists of $k_n$ identical tasks that can be executed in parallel, and that can be encoded into at least $k_n$ "replicas" of the same size (by introducing redundancy) so that the job is considered to be completed when any $k_n$ replicas associated with it finish their service. Moreover, we assume that servers can experience random slowdowns in their processing rate so that the service time of a replica is the product of its size and a random slowdown. First, we assume that the server slowdowns are shifted exponential and independent of the replica sizes. In this setting we show that the delay of a typical job is asymptotically minimized (as $n\to\infty$) when the number of replicas per task is a constant that only depends on the arrival rate $λ$, and on the expected slowdown of servers. Second, we introduce a new model for the server slowdowns in which larger tasks experience less variable slowdowns than smaller tasks. In this setting we show that, under the class of policies where all replicas start their service at the same time, the delay of a typical job is asymptotically minimized (as $n\to\infty$) when the number of replicas per task is made to depend on the actual size of the tasks being replicated, with smaller tasks being replicated more than larger tasks.

cs.PF↗

arXiv2018

Collaboratively Learning the Best Option on Graphs, Using Bounded Local Memory

We consider multi-armed bandit problems in social groups wherein each individual has bounded memory and shares the common goal of learning the best arm/option. We say an individual learns the best option if eventually (as $t\to \infty$) it pulls only the arm with the highest expected reward. While this goal is provably impossible for an isolated individual due to bounded memory, we show that, in social groups, this goal can be achieved easily with the aid of social persuasion (i.e., communication) as long as the communication networks/graphs satisfy some mild conditions. To deal with the interplay between the randomness in the rewards and in the social interaction, we employ the {\em mean-field approximation} method. Considering the possibility that the individuals in the networks may not be exchangeable when the communication networks are not cliques, we go beyond the classic mean-field techniques and apply a refined version of mean-field approximation: (1) Using coupling we show that, if the communication graph is connected and is either regular or has doubly-stochastic degree-weighted adjacency matrix, with probability $\to 1$ as the social group size $N \to \infty$, every individual in the social group learns the best option. (2) If the minimum degree of the graph diverges as $N \to \infty$, over an arbitrary but given finite time horizon, the sample paths describing the opinion evolutions of the individuals are asymptotically independent. In addition, the proportions of the population with different opinions converge to the unique solution of a system of ODEs. In the solution of the obtained ODEs, the proportion of the population holding the correct opinion converges to $1$ exponentially fast in time. Notably, our results hold even if the communication graphs are highly sparse.

cs.DC↗

arXiv2018

Collaboratively Learning the Best Option, Using Bounded Memory

We consider multi-armed bandit problems in social groups wherein each individual has bounded memory and shares the common goal of learning the best arm/option. We say an individual learns the best option if eventually (as $t \to \infty$) it pulls only the arm with the highest average reward. While this goal is provably impossible for an isolated individual, we show that, in social groups, this goal can be achieved easily with the aid of social persuasion, i.e., communication. Specifically, we study the learning dynamics wherein an individual sequentially decides on which arm to pull next based on not only its private reward feedback but also the suggestions provided by randomly chosen peers. Our learning dynamics are hard to analyze via explicit probabilistic calculations due to the stochastic dependency induced by social interaction. Instead, we employ the mean-field approximation method from statistical physics and we show: (1) With probability $\to 1$ as the social group size $N \to \infty $, every individual in the social group learns the best option. (2) Over an arbitrary finite time horizon $[0, T]$, with high probability (in $N$), the fraction of individuals that prefer the best option grows to 1 exponentially fast as $t$ increases ($t\in [0, T]$). A major innovation of our mean-filed analysis is a simple yet powerful technique to deal with absorbing states in the interchange of limits $N \to \infty$ and $t \to \infty $. The mean-field approximation method allows us to approximate the probabilistic sample paths of our learning dynamics by a deterministic and smooth trajectory that corresponds to the unique solution of a well-behaved system of ordinary differential equations (ODEs). Such an approximation is desired because the analysis of a system of ODEs is relatively easier than that of the original stochastic system.

cs.LG↗

arXiv2018

A lower bound on the queueing delay in resource constrained load balancing

We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $λn$, with $0<λ<1$, and are immediately dispatched to one of several queues associated with $n$ identical servers with unit processing rate. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate), in order to drive the expected queueing delay in steady-state of a typical job to zero, as $n$ increases. We develop a novel approach to show that, within a certain broad class of "symmetric" policies, every dispatching policy with a message rate of the order of $n$, and with a memory of the order of $\log n$ bits, results in an expected queueing delay which is bounded away from zero, uniformly as $n\to\infty$.

math.PR↗

arXiv2017

Delay, memory, and messaging tradeoffs in distributed service systems

We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate $λn$, with $0<λ<1$, and are immediately dispatched by a centralized dispatcher to one of $n$ First-In-First-Out queues associated with $n$ identical servers. The dispatcher is endowed with a finite memory, and with the ability to exchange messages with the servers. We propose and study a resource-constrained "pull-based" dispatching policy that involves two parameters: (i) the number of memory bits available at the dispatcher, and (ii) the average rate at which servers communicate with the dispatcher. We establish (using a fluid limit approach) that the asymptotic, as $n\to\infty$, expected queueing delay is zero when either (i) the number of memory bits grows logarithmically with $n$ and the message rate grows superlinearly with $n$, or (ii) the number of memory bits grows superlogarithmically with $n$ and the message rate is at least $λn$. Furthermore, when the number of memory bits grows only logarithmically with $n$ and the message rate is proportional to $n$, we obtain a closed-form expression for the (now positive) asymptotic delay. Finally, we demonstrate an interesting phase transition in the resource-constrained regime where the asymptotic delay is non-zero. In particular, we show that for any given $α>0$ (no matter how small), if our policy only uses a linear message rate $αn$, the resulting asymptotic delay is upper bounded, uniformly over all $λ<1$; this is in sharp contrast to the delay obtained when no messages are used ($α= 0$), which grows as $1/(1-λ)$ when $λ\uparrow 1$, or when the popular power-of-$d$-choices is used, in which the delay grows as $\log(1/(1-λ))$.

math.PR↗