arXiv subjects

Debankur Mukherjee

Publications and source records attributed to Debankur Mukherjee.

At least 19 recordsLinked to original sources

arXiv2026

Uniform-in-Time Boltzmann Mean-Field Limits for Anchored Binary Opinion Dynamics

We study a continuous-time opinion model in which agents interact in pairs and each agent has a fixed anchor. At each interaction epoch, both opinions are updated according to a possibly nonlinear rule with random inputs. Under suitable stability and moment assumptions, we establish uniform-in-time propagation of chaos: as the population grows, the joint law of any fixed number of agents converges, uniformly over time, to the corresponding product of the law solving a nonlinear Boltzmann equation. We first obtain a graphical estimate in total variation on finite time intervals, assuming only that the interaction rule is measurable. We extend this approximation uniformly over time by showing that both the finite system and its nonlinear limit converge exponentially fast to their respective stationary laws. We also prove uniform-in-time convergence in Wasserstein distance of order two under a contraction condition in mean square. For this result, we assume either a bounded state space or a drift condition that keeps moments of some order greater than two uniformly bounded. We examine two applications. For an anchored Friedkin-Johnsen model with random coefficients, we derive an affine stochastic recurrence for the stationary deviation of an opinion from its anchor. We use this recurrence to give conditions for sub-Gaussian or power-law tails and to characterize the exponent in the power-law case. We also study a model of biased assimilation, in which agents favor information that agrees with their current opinions. With the anchor and incoming evidence held at neutral values, we determine when an interaction moves an opinion farther from the neutral value, even if it starts close to it. Simulations illustrate the tail predictions and show how a single peak in the opinion distribution can split into two.

math.PR↗

arXiv2026

Scaling Limits of Constant-Stepsize SGD at Flat Minima

For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar $\sqrt{\alpha}$ scaling and a Gaussian limit as $\alpha\downarrow 0$. We show that this behavior changes fundamentally for convex objectives $H$ with flat minima and (sub)quadratic tails. More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize $\alpha$, we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an $\alpha$-dependent metric. When the minimizer $x_\star$ has local flatness exponent $m\ge2$, meaning that $\nabla^2 H(x)\asymp \lVert x-x_\star\rVert^{m-2} I_d$ as $x\to x_\star$, we obtain a contraction bound with factor $1-c\alpha^{m-1}$, where $c>0$ is a constant. This recovers the factor $1-c\alpha$ in the quadratic case $m=2$. We then analyze the small-stepsize scaling limit. We show that the invariant law concentrates on the scale $\alpha^{1/m}$ and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation $$ dY_t=-h_0(Y_t)\,dt+\Sigma^{1/2}\,dB_t , $$ where $h_0$ is the limiting drift at the minimizer and $\Sigma$ denotes the asymptotic covariance. This recovers the Gaussian limit when $m=2$ and gives generally non-Gaussian stationary limits in the flat case $m>2$. Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.

cs.LG↗

arXiv2026

Waning Immunity Fails to Restore a Positive Epidemic Threshold on Power-Law Networks

In a seminal work, Chatterjee and Durrett (2009) established that for the SIS epidemic process on random graphs with power-law degree distributions, the infection survives for an exponentially long time (in the network size) for any fixed, positive infection rate. Equivalently, the critical infection rate separating polynomial and exponential survival regimes is zero. In contrast, a substantial body of work in the physics literature conjectures, based primarily on numerical evidence and heuristic mean-field arguments, that introducing waning immunity (as in the SIRS process) yields a strictly positive critical infection rate on random graphs with power-law degrees; see, e.g., Pastor-Satorras et al. (2015), Ferreira et al. (2016), Silva et al. (2022). In particular, below this threshold, the epidemic is expected to persist only for a polynomial duration. A recent work by Friedrich et al. (2024) reinforces this perspective by proving polynomial survival for the SIRS process on star graphs, which is in contrast to the exponential survival in the SIS case that underpins Chatterjee and Durrett's arguments. In this paper, we disprove this conjecture and show that the epidemic threshold is also zero for the SIRS process on the configuration model with power-law degree distribution with exponent $\tau>2$. Our proof uncovers a novel bottleneck structure for the SIRS dynamics, which we term a "hierarchical star" of order 2, and show that it sustains the infection for an exponentially long time with high probability.

math.PR↗

arXiv2026

Higher-Order Approximations of Sojourn Times in M/G/1 Queues via Stein's Method

We study the stationary sojourn time distribution in an M/G/1 queue operating under heavy traffic. It is known that the sojourn time converges to an exponential distribution in the limit. Our focus is on obtaining pre-asymptotic, higher-order approximations that go beyond the classical exponential limit. Using Stein's method, we develop an approach based on higher-order expansions of the generator of the underlying Markov process. The key technical step is to represent higher-order derivatives in terms of lower-order ones and control the resulting error via derivative bounds of the Stein equation. Under suitable moment-matching conditions on the service distribution, we show that the approximation error decays as a high-order power of the slack parameter $\varepsilon=1-\rho$. Error bounds are established in the Zolotarev metric, which further imply bounds on the Wasserstein distance as well as the moments. Our results demonstrate that the accuracy of the exponential approximation can be systematically improved by matching progressively more moments of the service distribution.

math.PR↗

arXiv2026

SCaLE: Switching Cost aware Learning and Exploration

This work addresses the fundamental problem of unbounded metric movement costs in bandit online convex optimization, by considering high-dimensional dynamic quadratic hitting costs and $\ell_2$-norm switching costs in a noisy bandit feedback model. For a general class of stochastic environments, we provide the first algorithm SCaLE that provably achieves a distribution-agnostic sub-linear dynamic regret, without the knowledge of hitting cost structure. En-route, we present a novel spectral regret analysis that separately quantifies eigenvalue-error driven regret and eigenbasis-perturbation driven regret. Extensive numerical experiments, against online-learning baselines, corroborate our claims, and highlight statistical consistency of our algorithm.

cs.LG↗

arXiv2024

Optimal Decentralized Smoothed Online Convex Optimization

We study the multi-agent Smoothed Online Convex Optimization (SOCO) problem, where $N$ agents interact through a communication graph. In each round, each agent $i$ receives a strongly convex hitting cost function $f^i_t$ in an online fashion and selects an action $x^i_t \in \mathbb{R}^d$. The objective is to minimize the global cumulative cost, which includes the sum of individual hitting costs $f^i_t(x^i_t)$, a temporal "switching cost" for changing decisions, and a spatial "dissimilarity cost" that penalizes deviations in decisions among neighboring agents. We propose the first truly decentralized algorithm ACORD for multi-agent SOCO that provably exhibits asymptotic optimality. Our approach allows each agent to operate using only local information from its immediate neighbors in the graph. For finite-time performance, we establish that the optimality gap in the competitive ratio decreases with time horizon $T$ and can be conveniently tuned based on the per-round computation available to each agent. Our algorithm benefits from a provably scalable computational complexity that depends only logarithmically on the number of agents and almost linearly on their degree within the graph. Moreover, our results hold even when the communication graph changes arbitrarily and adaptively over time. Finally, ACORD, by virtue of its asymptotic-optimality, is shown to be provably superior to the state-of-the-art LPC algorithm, while exhibiting much lower computational complexity. Extensive numerical experiments across various network topologies further corroborate our theoretical claims.

math.OC↗

arXiv2024

Best of Both Worlds Guarantees for Smoothed Online Quadratic Optimization

We study the smoothed online quadratic optimization (SOQO) problem where, at each round $t$, a player plays an action $x_t$ in response to a quadratic hitting cost and an additional squared $\ell_2$-norm cost for switching actions. This problem class has strong connections to a wide range of application domains including smart grid management, adaptive control, and data center management, where switching-efficient algorithms are highly sought after. We study the SOQO problem in both adversarial and stochastic settings, and in this process, perform the first stochastic analysis of this class of problems. We provide the online optimal algorithm when the minimizers of the hitting cost function evolve as a general stochastic process, which, for the case of martingale process, takes the form of a distribution-agnostic dynamic interpolation algorithm (LAI). Next, we present the stochastic-adversarial trade-off by proving an $Ω(T)$ expected regret for the adversarial optimal algorithm in the literature (ROBD) with respect to LAI and, a sub-optimal competitive ratio for LAI in the adversarial setting. Finally, we present a best-of-both-worlds algorithm that obtains a robust adversarial performance while simultaneously achieving a near-optimal stochastic performance.

math.OC↗

arXiv2024

On $r$-to-$p$ norms of random matrices with nonnegative entries: Asymptotic normality and $\ell_\infty$-bounds for the maximizer

For an $n\times n$ matrix $A_n$, the $r\to p$ operator norm is defined as $$\|A_n\|_{r\to p}:= \sup_{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x} \|_r\leq 1 } \|A_n\mathbf{x} \|_p\quad\text{for}\quad r,p\geq 1.$$ For different choices of $r$ and $p$, this norm corresponds to key quantities that arise in diverse applications including matrix condition number estimation, clustering of data, and construction of oblivious routing schemes in transportation networks. This article considers $r\to p$ norms of symmetric random matrices with nonnegative entries, including adjacency matrices of Erdős-Rényi random graphs, matrices with positive sub-Gaussian entries, and certain sparse matrices. For $1<p\leq r<\infty$, the asymptotic normality, as $n\to\infty$, of the appropriately centered and scaled norm $\|A_n\|_{r\to p}$ is established. When $p \geq 2$, this is shown to imply asymptotic normality of the solution to the $\ell_p$ quadratic maximization problem, also known as the $\ell_p$ Grothendieck problem. Furthermore, a sharp $\ell_\infty$-approximation bound for the unique maximizing vector in the definition of $\|A_n\|_{r\to p}$ is obtained, and may be viewed as an $\ell_\infty$-stability result of the maximizer under random perturbations of the matrix with mean entries. This result is in fact shown to hold for a broad class of deterministic sequences of matrices having certain asymptotic expansion properties. The results obtained can be viewed as a generalization of the seminal results of Füredi and Komlós (1981) on asymptotic normality of the largest singular value of a class of symmetric random matrices. In the general case with $1<p\leq r< \infty$, spectral methods are no longer applicable, and so a new approach is developed involving a refined convergence analysis of a nonlinear power method and a perturbation bound on the maximizing vector, which may be of independent interest.

math.PR↗

arXiv2023

Self-Learning Threshold-Based Load Balancing

We consider a large-scale service system where incoming tasks have to be instantaneously dispatched to one out of many parallel server pools. The user-perceived performance degrades with the number of concurrent tasks and the dispatcher aims at maximizing the overall quality-of-service by balancing the load through a simple threshold policy. We demonstrate that such a policy is optimal on the fluid and diffusion scales, while only involving a small communication overhead, which is crucial for large-scale deployments. In order to set the threshold optimally, it is important, however, to learn the load of the system, which may be unknown. For that purpose, we design a control rule for tuning the threshold in an online manner. We derive conditions which guarantee that this adaptive threshold settles at the optimal value, along with estimates for the time until this happens. In addition, we provide numerical experiments which support the theoretical results and further indicate that our policy copes effectively with time-varying demand patterns.

cs.PF↗

arXiv2023

Optimal Rate-Matrix Pruning For Large-Scale Heterogeneous Systems

We present an analysis of large-scale load balancing systems, where the processing time distribution of tasks depends on both the task and server types. Our study focuses on the asymptotic regime, where the number of servers and task types tend to infinity in proportion. In heterogeneous environments, commonly used load balancing policies such as Join Fastest Idle Queue and Join Fastest Shortest Queue exhibit poor performance and even shrink the stability region. Interestingly, prior to this work, finding a scalable policy with a provable performance guarantee in this setup remained an open question. To address this gap, we propose and analyze two asymptotically delay-optimal dynamic load balancing policies. The first policy efficiently reserves the processing capacity of each server for ``good" tasks and routes tasks using the vanilla Join Idle Queue policy. The second policy, called the speed-priority policy, significantly increases the likelihood of assigning tasks to the respective ``good" servers capable of processing them at high speeds. By leveraging a framework inspired by the graphon literature and employing the mean-field method and stochastic coupling arguments, we demonstrate that both policies achieve asymptotic zero queuing. Specifically, as the system scales, the probability of a typical task being assigned to an idle server approaches 1.

math.PR↗

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

Mean-field Analysis for Load Balancing on Spatial Graphs

The analysis of large-scale, parallel-server load balancing systems has relied heavily on mean-field analysis. A pivotal assumption for this framework is that the servers are exchangeable. However, modern data-centers have data locality constraints, where tasks of a particular type can only be routed to a small subset of servers. An emerging line of research, therefore, considers load balancing algorithms on bipartite graphs where vertices in the two partitions represent the task types and servers, respectively, and an edge represents the server's ability to process the corresponding task type. Due to the lack of exchangeability in this model, the mean-field techniques fundamentally break down. Recent progress has been made by considering graphs with strong edge-expansion properties, i.e., where any two large subsets of vertices are well-connected. However, data locality often leads to spatial constraints, where edges are local. As a result, these bipartite graphs do not have strong expansion properties. In this paper, we consider the power-of-$d$ choices algorithm and develop a novel coupling-based approach to establish mean-field approximation for a large class of graphs that includes spatial graphs. As a corollary, we also show that, as the system size becomes large, the steady-state occupancy process for arbitrary regular bipartite graphs with diverging degrees, is indistinguishable from a fully flexible system on a complete bipartite graph. The method extends the scope of mean-field analysis far beyond the classical full-flexibility setup. En route, we prove that, starting from suitable states, the occupancy process becomes close to its steady state in a time that is independent of $N$. Such a large-scale mixing-time result might be of independent interest. Numerical experiments are conducted, which positively support the theoretical results.

math.PR↗

arXiv2022

Exploiting Data Locality to Improve Performance of Heterogeneous Server Clusters

We consider load balancing in large-scale heterogeneous server systems in the presence of data locality that imposes constraints on which tasks can be assigned to which servers. The constraints are naturally captured by a bipartite graph between the servers and the dispatchers handling assignments of various arrival flows. When a task arrives, the corresponding dispatcher assigns it to a server with the shortest queue among $d\geq 2$ randomly selected servers obeying the above constraints. Server processing speeds are heterogeneous and they depend on the server-type. For a broad class of bipartite graphs, we characterize the limit of the appropriately scaled occupancy process, both on the process-level and in steady state, as the system size becomes large. Using such a characterization, we show that data locality constraints can be used to significantly improve the performance of heterogeneous systems. This is in stark contrast to either heterogeneous servers in a full flexible system or data locality constraints in systems with homogeneous servers, both of which have been observed to degrade the system performance. Extensive numerical experiments corroborate the theoretical results.

math.PR↗

arXiv2022

Smoothed Online Optimization with Unreliable Predictions

We examine the problem of smoothed online optimization, where a decision maker must sequentially choose points in a normed vector space to minimize the sum of per-round, non-convex hitting costs and the costs of switching decisions between rounds. The decision maker has access to a black-box oracle, such as a machine learning model, that provides untrusted and potentially inaccurate predictions of the optimal decision in each round. The goal of the decision maker is to exploit the predictions if they are accurate, while guaranteeing performance that is not much worse than the hindsight optimal sequence of decisions, even when predictions are inaccurate. We impose the standard assumption that hitting costs are globally $α$-polyhedral. We propose a novel algorithm, Adaptive Online Switching (AOS), and prove that, for a large set of feasible $δ> 0$, it is $(1+δ)$-competitive if predictions are perfect, while also maintaining a uniformly bounded competitive ratio of $2^{\tilde{\mathcal{O}}(1/(αδ))}$ even when predictions are adversarial. Further, we prove that this trade-off is necessary and nearly optimal in the sense that \emph{any} deterministic algorithm which is $(1+δ)$-competitive if predictions are perfect must be at least $2^{\tildeΩ(1/(αδ))}$-competitive when predictions are inaccurate. In fact, we observe a unique threshold-type behavior in this trade-off: if $δ$ is not in the set of feasible options, then \emph{no} algorithm is simultaneously $(1 + δ)$-competitive if predictions are perfect and $ζ$-competitive when predictions are inaccurate for any $ζ< \infty$. Furthermore, we discuss that memory is crucial in AOS by proving that any algorithm that does not use memory cannot benefit from predictions. We complement our theoretical results by a numerical study on a microgrid application.

cs.LG↗

arXiv2022

Online Capacity Scaling Augmented With Unreliable Machine Learning Predictions

Modern data centers suffer from immense power consumption. As a result, data center operators have heavily invested in capacity scaling solutions, which dynamically deactivate servers if the demand is low and activate them again when the workload increases. We analyze a continuous-time model for capacity scaling, where the goal is to minimize the weighted sum of flow-time, switching cost, and power consumption in an online fashion. We propose a novel algorithm, called Adaptive Balanced Capacity Scaling (ABCS), that has access to black-box machine learning predictions. ABCS aims to adapt to the predictions and is also robust against unpredictable surges in the workload. In particular, we prove that ABCS is $(1+\varepsilon)$-competitive if the predictions are accurate, and yet, it has a uniformly bounded competitive ratio even if the predictions are completely inaccurate. Finally, we investigate the performance of this algorithm on a real-world dataset and carry out extensive numerical experiments, which positively support the theoretical results.

cs.DS↗

arXiv2021

Scalable load balancing in networked systems: A survey of recent advances

The basic load balancing scenario involves a single dispatcher where tasks arrive that must immediately be forwarded to one of $N$ single-server queues. We discuss recent advances on scalable load balancing schemes which provide favorable delay performance when $N$ grows large, and yet only require minimal implementation overhead. Join-the-Shortest-Queue (JSQ) yields vanishing delays as $N$ grows large, as in a centralized queueing arrangement, but involves a prohibitive communication burden. In contrast, power-of-$d$ or JSQ($d$) schemes that assign an incoming task to a server with the shortest queue among $d$ servers selected uniformly at random require little communication, but lead to constant delays. In order to examine this fundamental trade-off between delay performance and implementation overhead, we consider JSQ($d(N)$) schemes where the diversity parameter $d(N)$ depends on $N$ and investigate what growth rate of $d(N)$ is required to asymptotically match the optimal JSQ performance on fluid and diffusion scale. Stochastic coupling techniques and stochastic-process limits play an instrumental role in establishing the asymptotic optimality. We demonstrate how this methodology carries over to infinite-server settings, finite buffers, multiple dispatchers, servers arranged on graph topologies, and token-based load balancing including the popular Join-the-Idle-Queue (JIQ) scheme. In this way we provide a broad overview of the many recent advances in the field. This survey extends the short review presented at ICM 2018 (arXiv:1712.08555).

math.PR↗

arXiv2021

Many-server asymptotics for Join-the-Shortest Queue in the Super-Halfin-Whitt Scaling Window

The Join-the-Shortest Queue (JSQ) policy is a classical benchmark for the performance of many-server queueing systems due to its strong optimality properties. While the exact analysis of the JSQ policy is an open question to date, even under Markovian assumption on the service requirements, recently, there has been a significant progress in understanding its many-server asymptotic behavior since the work of Eschenfeldt and Gamarnik (Math.~Oper.~Res.~43 (2018) 867--886). In this paper, we analyze the many-server limits of the JSQ policy in the \emph{super-Halfin-Whitt} scaling window when load per server $\lambda_N$ scales with the system size $N$ as $\lim_{N\rightarrow\infty}N^{\alpha}(1-\lambda_N)=\beta$ for $\alpha\in (1/2, 1)$ and $\beta>0$. We establish that the centered and scaled total queue length process converges to a certain Bessel process with negative drift and the associated centered and scaled steady-state total queue length, indexed by $N$, converges to a $\mathrm{Gamma}(2,\beta)$ distribution. Both the transient and steady-state limit laws are universal in the sense that they do not depend on the value of the scaling parameter $\alpha$, and exhibit fundamentally different qualitative behavior from both the Halfin-Whitt regime ($\alpha = 1/2$) and the Non-degenerate Slowdown (NDS) regime ($\alpha=1$).

math.PR↗

arXiv2020

Load Balancing Under Strict Compatibility Constraints

We study large-scale systems operating under the JSQ$(d)$ policy in the presence of stringent task-server compatibility constraints. Consider a system with $N$ identical single-server queues and $M(N)$ task types, where each server is able to process only a small subset of possible task types. Each arriving task selects $d\geq 2$ random servers compatible to its type, and joins the shortest queue among them. The compatibility constraint is naturally captured by a fixed bipartite graph $G_N$ between the servers and the task types. When $G_N$ is complete bipartite, the meanfield approximation is proven to be accurate. However, such dense compatibility graphs are infeasible due to their overwhelming implementation cost and prohibitive storage capacity requirement at the servers. Our goal in this paper is to characterize the class of sparse compatibility graphs for which the meanfield approximation remains valid. To achieve this, first, we introduce a novel graph expansion-based notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs match the performance of a fully flexible system, asymptotically in the large-system limit. Furthermore, for any $c(N)$ satisfying $$\frac{Nc(N)}{M(N)\ln(N)}\to \infty\quad \text{and}\quad c(N)\to \infty,$$ as $N\to\infty$, we show that proportionally sparse random compatibility graphs can be designed, so that the degree of each server is at most $c(N)$. This reduces the server-degree almost by a factor $N/\ln(N)$, compared to the complete bipartite compatibility graph, while maintaining the same asymptotic performance. Extensive simulation experiments are conducted to corroborate the theoretical results.

math.PR↗