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Siva Theja Maguluri

Publications and source records attributed to Siva Theja Maguluri.

At least 37 records · Page 2Linked to original sources
arXiv2025

A Non-Asymptotic Theory of Seminorm Lyapunov Stability: From Deterministic to Stochastic Iterative Algorithms

We study the problem of solving fixed-point equations for seminorm-contractive operators and establish foundational results on the non-asymptotic behavior of iterative algorithms in both deterministic and stochastic settings. Specifically, in the deterministic setting, we prove a fixed-point theorem for seminorm-contractive operators, showing that iterates converge geometrically to the kernel of the seminorm. In the stochastic setting, we analyze the corresponding stochastic approximation (SA) algorithm under seminorm-contractive operators and Markovian noise, providing a finite-sample analysis for various stepsize choices. A benchmark for equation solving is linear systems of equations, where the convergence behavior of fixed-point iteration is closely tied to the stability of linear dynamical systems. In this special case, our results provide a complete characterization of system stability with respect to a seminorm, linking it to the solution of a Lyapunov equation in terms of positive semi-definite matrices. In the stochastic setting, we establish a finite-sample analysis for linear Markovian SA without requiring the Hurwitzness assumption. Our theoretical results offer a unified framework for deriving finite-sample bounds for various reinforcement learning algorithms in the average reward setting, including TD($λ$) for policy evaluation (which is a special case of solving a Poisson equation) and Q-learning for control.

cs.LG↗
arXiv2024

Federated Stochastic Approximation under Markov Noise and Heterogeneity: Applications in Reinforcement Learning

Since reinforcement learning algorithms are notoriously data-intensive, the task of sampling observations from the environment is usually split across multiple agents. However, transferring these observations from the agents to a central location can be prohibitively expensive in terms of communication cost, and it can also compromise the privacy of each agent's local behavior policy. Federated reinforcement learning is a framework in which $N$ agents collaboratively learn a global model, without sharing their individual data and policies. This global model is the unique fixed point of the average of $N$ local operators, corresponding to the $N$ agents. Each agent maintains a local copy of the global model and updates it using locally sampled data. In this paper, we show that by careful collaboration of the agents in solving this joint fixed point problem, we can find the global model $N$ times faster, also known as linear speedup. We first propose a general framework for federated stochastic approximation with Markovian noise and heterogeneity, showing linear speedup in convergence. We then apply this framework to federated reinforcement learning algorithms, examining the convergence of federated on-policy TD, off-policy TD, and $Q$-learning.

cs.LG↗
arXiv2024

Markov Chain Variance Estimation: A Stochastic Approximation Approach

We consider the problem of estimating the asymptotic variance of a function defined on a Markov chain, an important step for statistical inference of the stationary mean. We design a novel recursive estimator that requires $O(1)$ computation at each step, does not require storing any historical samples or any prior knowledge of run-length, and has optimal $O(\frac{1}{n})$ rate of convergence for the mean-squared error (MSE) with provable finite sample guarantees. Here, $n$ refers to the total number of samples generated. Our estimator is based on linear stochastic approximation of an equivalent formulation of the asymptotic variance in terms of the solution of the Poisson equation. We generalize our estimator in several directions, including estimating the covariance matrix for vector-valued functions, estimating the stationary variance of a Markov chain, and approximately estimating the asymptotic variance in settings where the state space of the underlying Markov chain is large. We also show applications of our estimator in average reward reinforcement learning (RL), where we work with asymptotic variance as a risk measure to model safety-critical applications. We design a temporal-difference type algorithm tailored for policy evaluation in this context. We consider both the tabular and linear function approximation settings. Our work paves the way for developing actor-critic style algorithms for variance-constrained RL.

math.ST↗
arXiv2024

Performance of NPG in Countable State-Space Average-Cost RL

We consider policy optimization methods in reinforcement learning settings where the state space is arbitrarily large, or even countably infinite. The motivation arises from control problems in communication networks, matching markets, and other queueing systems. Specifically, we consider the popular Natural Policy Gradient (NPG) algorithm, which has been studied in the past only under the assumption that the cost is bounded and the state space is finite, neither of which holds for the aforementioned control problems. Assuming a Lyapunov drift condition, which is naturally satisfied in some cases and can be satisfied in other cases at a small cost in performance, we design a state-dependent step-size rule which dramatically improves the performance of NPG for our intended applications. In addition to experimentally verifying the performance improvement, we also theoretically show that the iteration complexity of NPG can be made independent of the size of the state space. The key analytical tool we use is the connection between NPG step-sizes and the solution to Poisson's equation. In particular, we provide policy-independent bounds on the solution to Poisson's equation, which are then used to guide the choice of NPG step-sizes.

cs.LG↗
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

Heavy Traffic Queue Length Behaviour in a Switch under Markovian Arrivals

This paper studies the input queued switch operating under the MaxWeight algorithm when the arrivals are according to a Markovian process. We exactly characterize the heavy-traffic scaled mean sum queue length in the heavy-traffic limit, and shows that it is within a factor of less than $2$ from a universal lower bound. Moreover, we obtain lower and upper bounds, that are applicable in all traffic regimes, and they become tight in the heavy-traffic regime. The paper obtains these results by generalizing the drift method recently developed for the case of i.i.d. arrivals, to the case of Markovian arrivals. The paper illustrates this generalization by first obtaining the heavy-traffic mean queue length and its distribution in a single server queue under Markovian arrivals and then applying it to the case of input queued switch. The key idea is to exploit the geometric mixing of finite-state Markov chains, and to work with a time horizon that is picked so that the error due to mixing depends on the heavy-traffic parameter.

math.PR↗
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

Heavy Traffic Joint Queue Length Distribution without Resource Pooling

This paper studies the heavy-traffic joint distribution of queue lengths in two stochastic processing networks (SPN), viz., an input-queued switch operating under the MaxWeight scheduling policy and a two-server parallel server system called the $\mathcal{N}$-system. These two systems serve as representatives of SPNs that do not satisfy the so-called Complete Resource Pooling (CRP) condition, and consequently exhibit a multidimensional State Space Collapse (SSC). Except in special cases, only mean queue lengths of such non-CRP systems is known in the literature. In this paper, we develop the Transform method to study the joint distribution of queue lengths in non-CRP systems. The key challenge is in solving an implicit functional equation involving the Laplace transform of the heavy-traffic limiting distribution. For the $\mathcal{N}$-system and a special case of an input-queued switch involving only three queues, we obtain the exact limiting heavy-traffic joint distribution in terms of a linear combination of two iid exponentials. For the general $n\times n$ input-queued switch that has $n^2$ queues, under a conjecture on uniqueness of the solution of the functional equation, we obtain an exact joint distribution of the heavy-traffic limiting queue-lengths in terms of a non-linear combination of $2n$ iid exponentials.

math.PR↗
arXiv2023

A Lyapunov Theory for Finite-Sample Guarantees of Asynchronous Q-Learning and TD-Learning Variants

This paper develops an unified framework to study finite-sample convergence guarantees of a large class of value-based asynchronous reinforcement learning (RL) algorithms. We do this by first reformulating the RL algorithms as \textit{Markovian Stochastic Approximation} (SA) algorithms to solve fixed-point equations. We then develop a Lyapunov analysis and derive mean-square error bounds on the convergence of the Markovian SA. Based on this result, we establish finite-sample mean-square convergence bounds for asynchronous RL algorithms such as $Q$-learning, $n$-step TD, TD$(λ)$, and off-policy TD algorithms including V-trace. As a by-product, by analyzing the convergence bounds of $n$-step TD and TD$(λ)$, we provide theoretical insights into the bias-variance trade-off, i.e., efficiency of bootstrapping in RL. This was first posed as an open problem in (Sutton, 1999).

cs.LG↗
arXiv2023

Exponential Tail Bounds on Queues: A Confluence of Non-Asymptotic Heavy Traffic and Large Deviations

In general, obtaining the exact steady-state distribution of queue lengths is not feasible. Therefore, we establish bounds for the tail probabilities of queue lengths. Specifically, we examine queueing systems under Heavy-Traffic (HT) conditions and provide exponentially decaying bounds for the probability $\mathbb P(εq > x)$, where $ε$ is the HT parameter denoting how far the load is from the maximum allowed load. Our bounds are not limited to asymptotic cases and are applicable even for finite values of $ε$, and they get sharper as $ε\to 0$. Consequently, we derive non-asymptotic convergence rates for the tail probabilities. Unlike other approaches such as moment bounds based on drift arguments and bounds on Wasserstein distance using Stein's method, our method yields sharper tail bounds. Furthermore, our results offer bounds on the exponential rate of decay of the tail, given by $-\frac{1}{x} \log \mathbb P(εq > x)$ for any finite value of $x$. These can be interpreted as non-asymptotic versions of Large Deviation (LD) results. We demonstrate our approach by presenting tail bounds for: (i) a continuous time Join-the-shortest queue (JSQ) load balancing system, (ii) a discrete time single-server queue and (iii) an $M/M/n$ queue. We not only bridge the gap between classical-HT and LD regimes but also explore the large system HT regimes for JSQ and $M/M/n$ systems. In these regimes, both the system size and the system load increase simultaneously. Our results also close a gap in the existing literature on the limiting distribution of JSQ in the super-NDS (a.k.a. super slowdown) regime. This contribution is of an independent interest. Here, a key ingredient is a more refined characterization of state space collapse for JSQ system, achieved by using an exponential Lyapunov function designed to approximate the $\ell_{\infty}$ norm.

math.PR↗
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↗
arXiv2023

Optimal Weighted-Delay Scheduling in $2\times 2$ Input-Queued Switches

Motivated by few delay-optimal scheduling results, in comparison to results on throughput optimality, we investigate a canonical input-queued switch scheduling problem in which the objective is to minimize the discounted delay cost over an infinite time horizon. We derive an optimal scheduling policy and establish corresponding theoretical properties, which are expected to be of interest more broadly than input-queued switches. Computational experiments demonstrate and quantify the benefits of our optimal scheduling policy over alternative policies such as variants of MaxWeight scheduling, well-known to be throughput optimal and more recently shown to be delay optimal in the heavy-traffic regime limit.

math.OC↗
arXiv2023

An Approximate Policy Iteration Viewpoint of Actor-Critic Algorithms

In this work, we consider policy-based methods for solving the reinforcement learning problem, and establish the sample complexity guarantees. A policy-based algorithm typically consists of an actor and a critic. We consider using various policy update rules for the actor, including the celebrated natural policy gradient. In contrast to the gradient ascent approach taken in the literature, we view natural policy gradient as an approximate way of implementing policy iteration, and show that natural policy gradient (without any regularization) enjoys geometric convergence when using increasing stepsizes. As for the critic, we consider using TD-learning with linear function approximation and off-policy sampling. Since it is well-known that in this setting TD-learning can be unstable, we propose a stable generic algorithm (including two specific algorithms: the $λ$-averaged $Q$-trace and the two-sided $Q$-trace) that uses multi-step return and generalized importance sampling factors, and provide the finite-sample analysis. Combining the geometric convergence of the actor with the finite-sample analysis of the critic, we establish for the first time an overall $\mathcal{O}(ε^{-2})$ sample complexity for finding an optimal policy (up to a function approximation error) using policy-based methods under off-policy sampling and linear function approximation.

cs.LG↗
arXiv2023

Transportation Polytope and its Applications in Parallel Server Systems

A parallel server system is a stochastic processing network with applications in manufacturing, supply chain, ride-hailing, call centers, etc. Heterogeneous customers arrive in the system, and only a subset of servers can serve any customer type given by the flexibility graph. The goal of the system operator is to minimize the delay that depends on the scheduling policy and the flexibility graph. A long line of literature focuses on designing near-optimal scheduling policies given a flexibility graph. On the contrary, we fix the scheduling policy to be the so-called MaxWeight scheduling given its superior delay performance and focus on designing near-optimal, sparse flexibility graphs. Our contributions are threefold. First, we analyze the expected delay in the heavy-traffic asymptotic regime in terms of the properties of the flexibility graph and use this result to translate the design question in terms of transportation polytope, the deterministic equivalent of parallel server queues. Second, we design the sparsest flexibility graph that achieves a given delay performance and shows the robustness of the design to demand uncertainty. Third, given the budget to add edges arrives sequentially in time, we present the optimal schedule for adding them to the flexibility graph. These results are obtained by proving new results for transportation polytopes and are of independent interest. In particular, translating the difficulties to a simpler model, i.e. transportation polytope, allows us to develop a unified framework to answer several design questions.

cs.NI↗
arXiv2022

Target Network and Truncation Overcome The Deadly Triad in $Q$-Learning

$Q$-learning with function approximation is one of the most empirically successful while theoretically mysterious reinforcement learning (RL) algorithms, and was identified in Sutton (1999) as one of the most important theoretical open problems in the RL community. Even in the basic linear function approximation setting, there are well-known divergent examples. In this work, we show that \textit{target network} and \textit{truncation} together are enough to provably stabilize $Q$-learning with linear function approximation, and we establish the finite-sample guarantees. The result implies an $O(ε^{-2})$ sample complexity up to a function approximation error. Moreover, our results do not require strong assumptions or modifying the problem parameters as in existing literature.

cs.LG↗
arXiv2022

Finite-Sample Analysis of Off-Policy Natural Actor-Critic with Linear Function Approximation

In this paper, we develop a novel variant of off-policy natural actor-critic algorithm with linear function approximation and we establish a sample complexity of $\mathcal{O}(ε^{-3})$, outperforming all the previously known convergence bounds of such algorithms. In order to overcome the divergence due to deadly triad in off-policy policy evaluation under function approximation, we develop a critic that employs $n$-step TD-learning algorithm with a properly chosen $n$. We present finite-sample convergence bounds on this critic under both constant and diminishing step sizes, which are of independent interest. Furthermore, we develop a variant of natural policy gradient under function approximation, with an improved convergence rate of $\mathcal{O}(1/T)$ after $T$ iterations. Combining the finite sample error bounds of actor and the critic, we obtain the $\mathcal{O}(ε^{-3})$ sample complexity. We derive our sample complexity bounds solely based on the assumption that the behavior policy sufficiently explores all the states and actions, which is a much lighter assumption compared to the related literature.

cs.LG↗
arXiv2022

Finite Sample Analysis of Two-Time-Scale Natural Actor-Critic Algorithm

Actor-critic style two-time-scale algorithms are one of the most popular methods in reinforcement learning, and have seen great empirical success. However, their performance is not completely understood theoretically. In this paper, we characterize the \emph{global} convergence of an online natural actor-critic algorithm in the tabular setting using a single trajectory of samples. Our analysis applies to very general settings, as we only assume ergodicity of the underlying Markov decision process. In order to ensure enough exploration, we employ an $ε$-greedy sampling of the trajectory. For a fixed and small enough exploration parameter $ε$, we show that the two-time-scale natural actor-critic algorithm has a rate of convergence of $\tilde{\mathcal{O}}(1/T^{1/4})$, where $T$ is the number of samples, and this leads to a sample complexity of $\Tilde{\mathcal{O}}(1/δ^{8})$ samples to find a policy that is within an error of $δ$ from the \emph{global optimum}. Moreover, by carefully decreasing the exploration parameter $ε$ as the iterations proceed, we present an improved sample complexity of $\Tilde{\mathcal{O}}(1/δ^{6})$ for convergence to the global optimum.

cs.LG↗
arXiv2022

Finite-Sample Analysis of Nonlinear Stochastic Approximation with Applications in Reinforcement Learning

Motivated by applications in reinforcement learning (RL), we study a nonlinear stochastic approximation (SA) algorithm under Markovian noise, and establish its finite-sample convergence bounds under various stepsizes. Specifically, we show that when using constant stepsize (i.e., $α_k\equiv α$), the algorithm achieves exponential fast convergence to a neighborhood (with radius $O(α\log(1/α))$) around the desired limit point. When using diminishing stepsizes with appropriate decay rate, the algorithm converges with rate $O(\log(k)/k)$. Our proof is based on Lyapunov drift arguments, and to handle the Markovian noise, we exploit the fast mixing of the underlying Markov chain. To demonstrate the generality of our theoretical results on Markovian SA, we use it to derive the finite-sample bounds of the popular $Q$-learning with linear function approximation algorithm, under a condition on the behavior policy. Importantly, we do not need to make the assumption that the samples are i.i.d., and do not require an artificial projection step in the algorithm to maintain the boundedness of the iterates. Numerical simulations corroborate our theoretical results.

math.OC↗