Siva Theja Maguluri

Shaan Ul Haque, Zedong Wang, Zixuan Zhang et al.

Stochastic approximation (SA) is a method for finding the root of an operator perturbed by noise. The focus of this paper is studying the distribution of SA iterates in finite time. In general, it is not possible to characterize the exact distribution, and therefore our goal is to find an approximation which can yield useful tail bounds. Inspired by the rich literature on the asymptotic normality of rescaled SA iterates, we approximate the pre-limit distributions by a sequence of Gaussians whose covariance is recursively defined. In particular, we establish explicit bounds on the Wasserstein-1 distance between the rescaled iterate at time $k$ and the aforementioned Gaussian for various choices of step-sizes. Since these covariances converge to the classical asymptotic limit, our analysis also provides a convergence rate for asymptotic normality as a by-product. As an immediate consequence of our bounds, we obtain tail bounds on the error of SA iterates at any time. Finally, we establish the sharpness of our rates by providing matching lower bounds and validate our findings through simulations. We obtain the sharp rates by first studying the convergence rate of the discrete Ornstein-Uhlenbeck (O-U) process driven by general noise, whose stationary distribution is identical to the limiting Gaussian distribution of the rescaled SA iterates. We believe that this is of independent interest, given its connection to sampling literature. The analysis involves adapting Stein's method for Gaussian approximation to handle the matrix weighted sum of i.i.d. random variables. The desired finite-time bounds for SA are obtained by characterizing the error dynamics between the rescaled SA iterate and the discrete time O-U process and combining it with the convergence rate of the latter process.

Zaiwei Chen, Siva Theja Maguluri

We survey Lyapunov-based techniques for the finite-time analysis of stochastic iterative algorithms, also known as stochastic approximation (SA) algorithms, for solving fixed-point equations $\bar{F}(x)=x$, where the operator $\bar{F}(\cdot)$ can only be accessed through a noisy oracle. We first focus on the standard setting in which $\bar{F}(\cdot)$ is contractive with respect to some norm and the noise is i.i.d., and explain how generalized Moreau envelopes serve as universal Lyapunov functions, regardless of the underlying norm. We then show how this framework yields mean-square convergence guarantees and applies to stochastic gradient descent, linear SA, and value-based reinforcement learning algorithms such as Q-learning and temporal-difference learning. Finally, we discuss extensions to Markovian noise, seminorm-contractive operators, dissipative operators, and high-probability bounds, and conclude with open problems. The goal is to present a unified and self-contained roadmap for the finite-time analysis of SA and its applications, especially in reinforcement learning.

Sajad Khodadadian, Pranay Sharma, Gauri Joshi et al.

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.

Zaiwei Chen, John Paul Clarke, Siva Theja Maguluri

$Q$-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.

Zaiwei Chen, Siva Theja Maguluri, Martin Zubeldia

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.

Zaiwei Chen, Siva Theja Maguluri

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.

Zedong Wang, Siva Theja Maguluri

We study a heavily overloaded single-server queue with abandonment and derive bounds on stationary tail probabilities of the queue length. As the abandonment rate $γ\downarrow 0$, the centered-scaled queue length $\tilde{q}$ is known to converge in distribution to a Gaussian. However, such asymptotic limits do not quantify the pre-limit tail $\mathbb{P}(\tilde{q}>a)$ for fixed $γ>0$. Our goal is to obtain pre-limit bounds that are \emph{efficient} across different deviation regimes. For constant deviations, efficiency means Gaussian-type decay in $a$ together with a pre-limit error that vanishes as $γ\downarrow 0$, yielding the correct Gaussian tail in the limit. We establish such an efficient bound that is best-of-both-worlds. For larger deviations when $a$ is a function of $γ$, efficiency translates into exponentially tight, matching upper and lower bounds. For moderate deviation, we obtain sub-Gaussian tails, while in the large deviation regime the decay becomes sub-Poisson. Our bounds are obtained using a combination of Stein's method for Wasserstein-$p$ distance and the transform method. We then consider a load-balancing system of abandonment queues with heterogeneous servers, operating under the join-the-shortest-queue (JSQ) policy in the heavily overloaded regime. As in the case of single-server queue, we again obtain Wasserstein-$p$ bounds w.r.t.\ a Gaussian, and efficient concentration for constant and moderate deviations. For larger deviations, our JSQ upper bounds exhibit a transition from Gaussian-type decay to sub-Weibull decay. All these results are obtained using Stein's method. In addition, a key ingredient here is establishing a state space collapse (SSC) where all queues become equal. We establish a $p$-th moment bound on the orthogonal component of the queue length vector that is essential for our Wasserstein-$p$ bound.

Shubhada Agrawal, Siva Theja Maguluri, Martin Zubeldia

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.

Shaan Ul Haque, Sajad Khodadadian, Siva Theja Maguluri

Stochastic approximation (SA) is an iterative algorithm for finding the fixed point of an operator using noisy samples and widely used in optimization and Reinforcement Learning (RL). The noise in RL exhibits a Markovian structure, and in some cases, such as gradient temporal difference (GTD) methods, SA is employed in a two-time-scale framework. This combination introduces significant theoretical challenges for analysis. We derive an upper bound on the error for the iterations of linear two-time-scale SA with Markovian noise. We demonstrate that the mean squared error decreases as $trace (Σ^y)/k + o(1/k)$ where $k$ is the number of iterates, and $Σ^y$ is an appropriately defined covariance matrix. A key feature of our bounds is that the leading term, $Σ^y$, exactly matches with the covariance in the Central Limit Theorem (CLT) for the two-time-scale SA, and we call them tight finite-time bounds. We illustrate their use in RL by establishing sample complexity for off-policy algorithms, TDC, GTD, and GTD2. A special case of linear two-time-scale SA that is extensively studied is linear SA with Polyak-Ruppert averaging. We present tight finite time bounds corresponding to the covariance matrix of the CLT. Such bounds can be used to study TD-learning with Polyak-Ruppert averaging.

Zaiwei Chen, Sheng Zhang, Zhe Zhang et al.

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.

Shaan Ul Haque, Siva Theja Maguluri

Motivated by engineering applications such as resource allocation in networks and inventory systems, we consider average-reward Reinforcement Learning with unbounded state space and reward function. Recent works studied this problem in the actor-critic framework and established finite sample bounds assuming access to a critic with certain error guarantees. We complement their work by studying Temporal Difference (TD) learning with linear function approximation and establishing finite-time bounds with the optimal $\mathcal{O}\left(1/ε^2\right)$ sample complexity. These results are obtained using the following general-purpose theorem for non-linear Stochastic Approximation (SA). Suppose that one constructs a Lyapunov function for a non-linear SA with certain drift condition. Then, our theorem establishes finite-time bounds when this SA is driven by unbounded Markovian noise under suitable conditions. It serves as a black box tool to generalize sample guarantees on SA from i.i.d. or martingale difference case to potentially unbounded Markovian noise. The generality and the mild assumption of the setup enables broad applicability of our theorem. We illustrate its power by studying two more systems: (i) We improve upon the finite-time bounds of $Q$-learning by tightening the error bounds and also allowing for a larger class of behavior policies. (ii) We establish the first ever finite-time bounds for distributed stochastic optimization of high-dimensional smooth strongly convex function using cyclic block coordinate descent.

Zedong Wang, Yuyang Wang, Ijay Narang et al.

Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency. For a fixed stepsize, the iterates typically admit a stationary distribution that is rarely tractable. Prior work shows that as the stepsize $α\downarrow 0$, the centered-and-scaled steady state converges weakly to a Gaussian random vector. However, for fixed $α$, this weak convergence offers no usable error bound for approximating the steady-state by its Gaussian limit. This paper provides explicit, non-asymptotic error bounds for fixed $α$. We first prove general-purpose theorems that bound the Wasserstein distance between the centered-scaled steady state and an appropriate Gaussian distribution, under regularity conditions for drift and moment conditions for noise. To ensure broad applicability, we cover both i.i.d. and Markovian noise models. We then instantiate these theorems for three representative SA settings: (1) stochastic gradient descent (SGD) for smooth strongly convex objectives, (2) linear SA, and (3) contractive nonlinear SA. We obtain dimension- and stepsize-dependent, explicit bounds in Wasserstein distance of order $α^{1/2}\log(1/α)$ for small $α$. Building on the Wasserstein approximation error, we further derive non-uniform Berry--Esseen-type tail bounds that compare the steady-state tail probability to Gaussian tails. We achieve an explicit error term that decays in both the deviation level and stepsize $α$. We adapt the same analysis for SGD beyond strongly convexity and study general convex objectives. We identify a non-Gaussian (Gibbs) limiting law under the correct scaling, which is validated numerically, and provide a corresponding pre-limit Wasserstein error bound.

Isaac Grosof, Siva Theja Maguluri, R. Srikant

A wide variety of queueing systems can be naturally modeled as infinite-state Markov Decision Processes (MDPs). In the reinforcement learning (RL) context, a variety of algorithms have been developed to learn and optimize these MDPs. At the heart of many popular policy-gradient based learning algorithms, such as natural actor-critic, TRPO, and PPO, lies the Natural Policy Gradient (NPG) policy optimization algorithm. Convergence results for these RL algorithms rest on convergence results for the NPG algorithm. However, all existing results on the convergence of the NPG algorithm are limited to finite-state settings. We study a general class of queueing MDPs, and prove a $O(1/\sqrt{T})$ convergence rate for the NPG algorithm, if the NPG algorithm is initialized with the MaxWeight policy. This is the first convergence rate bound for the NPG algorithm for a general class of infinite-state average-reward MDPs. Moreover, our result applies to a beyond the queueing setting to any countably-infinite MDP satisfying certain mild structural assumptions, given a sufficiently good initial policy. Key to our result are state-dependent bounds on the relative value function achieved by the iterate policies of the NPG algorithm.

Zaiwei Chen, Shancong Mou, Siva Theja Maguluri

Stochastic approximation (SA) and stochastic gradient descent (SGD) algorithms are work-horses for modern machine learning algorithms. Their constant stepsize variants are preferred in practice due to fast convergence behavior. However, constant step stochastic iterative algorithms do not converge asymptotically to the optimal solution, but instead have a stationary distribution, which in general cannot be analytically characterized. In this work, we study the asymptotic behavior of the appropriately scaled stationary distribution, in the limit when the constant stepsize goes to zero. Specifically, we consider the following three settings: (1) SGD algorithms with smooth and strongly convex objective, (2) linear SA algorithms involving a Hurwitz matrix, and (3) nonlinear SA algorithms involving a contractive operator. When the iterate is scaled by $1/\sqrtα$, where $α$ is the constant stepsize, we show that the limiting scaled stationary distribution is a solution of an integral equation. Under a uniqueness assumption (which can be removed in certain settings) on this equation, we further characterize the limiting distribution as a Gaussian distribution whose covariance matrix is the unique solution of a suitable Lyapunov equation. For SA algorithms beyond these cases, our numerical experiments suggest that unlike central limit theorem type results: (1) the scaling factor need not be $1/\sqrtα$, and (2) the limiting distribution need not be Gaussian. Based on the numerical study, we come up with a formula to determine the right scaling factor, and make insightful connection to the Euler-Maruyama discretization scheme for approximating stochastic differential equations.

Zaiwei Chen, Siva Theja Maguluri, Sanjay Shakkottai et al.

In temporal difference (TD) learning, off-policy sampling is known to be more practical than on-policy sampling, and by decoupling learning from data collection, it enables data reuse. It is known that policy evaluation (including multi-step off-policy importance sampling) has the interpretation of solving a generalized Bellman equation. In this paper, we derive finite-sample bounds for any general off-policy TD-like stochastic approximation algorithm that solves for the fixed-point of this generalized Bellman operator. Our key step is to show that the generalized Bellman operator is simultaneously a contraction mapping with respect to a weighted $\ell_p$-norm for each $p$ in $[1,\infty)$, with a common contraction factor. Off-policy TD-learning is known to suffer from high variance due to the product of importance sampling ratios. A number of algorithms (e.g. $Q^π(λ)$, Tree-Backup$(λ)$, Retrace$(λ)$, and $Q$-trace) have been proposed in the literature to address this issue. Our results immediately imply finite-sample bounds of these algorithms. In particular, we provide first-known finite-sample guarantees for $Q^π(λ)$, Tree-Backup$(λ)$, and Retrace$(λ)$, and improve the best known bounds of $Q$-trace in [19]. Moreover, we show the bias-variance trade-offs in each of these algorithms.

Zaiwei Chen, Sajad Khodadadian, Siva Theja Maguluri

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.

Sajad Khodadadian, Prakirt Raj Jhunjhunwala, Sushil Mahavir Varma et al.

Markov Decision Processes are classically solved using Value Iteration and Policy Iteration algorithms. Recent interest in Reinforcement Learning has motivated the study of methods inspired by optimization, such as gradient ascent. Among these, a popular algorithm is the Natural Policy Gradient, which is a mirror descent variant for MDPs. This algorithm forms the basis of several popular Reinforcement Learning algorithms such as Natural actor-critic, TRPO, PPO, etc, and so is being studied with growing interest. It has been shown that Natural Policy Gradient with constant step size converges with a sublinear rate of O(1/k) to the global optimal. In this paper, we present improved finite time convergence bounds, and show that this algorithm has geometric (also known as linear) asymptotic convergence rate. We further improve this convergence result by introducing a variant of Natural Policy Gradient with adaptive step sizes. Finally, we compare different variants of policy gradient methods experimentally.

Sajad Khodadadian, Zaiwei Chen, Siva Theja Maguluri

In this paper, we provide finite-sample convergence guarantees for an off-policy variant of the natural actor-critic (NAC) algorithm based on Importance Sampling. In particular, we show that the algorithm converges to a global optimal policy with a sample complexity of $\mathcal{O}(ε^{-3}\log^2(1/ε))$ under an appropriate choice of stepsizes. In order to overcome the issue of large variance due to Importance Sampling, we propose the $Q$-trace algorithm for the critic, which is inspired by the V-trace algorithm \cite{espeholt2018impala}. This enables us to explicitly control the bias and variance, and characterize the trade-off between them. As an advantage of off-policy sampling, a major feature of our result is that we do not need any additional assumptions, beyond the ergodicity of the Markov chain induced by the behavior policy.

Zaiwei Chen, Siva Theja Maguluri, Sanjay Shakkottai et al.

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).

Sajad Khodadadian, Thinh T. Doan, Justin Romberg et al.

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.

Zaiwei Chen, Siva Theja Maguluri, Sanjay Shakkottai et al.

Stochastic Approximation (SA) is a popular approach for solving fixed-point equations where the information is corrupted by noise. In this paper, we consider an SA involving a contraction mapping with respect to an arbitrary norm, and show its finite-sample error bounds while using different stepsizes. The idea is to construct a smooth Lyapunov function using the generalized Moreau envelope, and show that the iterates of SA have negative drift with respect to that Lyapunov function. Our result is applicable in Reinforcement Learning (RL). In particular, we use it to establish the first-known convergence rate of the V-trace algorithm for off-policy TD-learning. Moreover, we also use it to study TD-learning in the on-policy setting, and recover the existing state-of-the-art results for $Q$-learning. Importantly, our construction results in only a logarithmic dependence of the convergence bound on the size of the state-space.

Thinh T. Doan, Siva Theja Maguluri, Justin Romberg

We study the policy evaluation problem in multi-agent reinforcement learning, modeled by a Markov decision process. In this problem, the agents operate in a common environment under a fixed control policy, working together to discover the value (global discounted accumulative reward) associated with each environmental state. Over a series of time steps, the agents act, get rewarded, update their local estimate of the value function, then communicate with their neighbors. The local update at each agent can be interpreted as a distributed variant of the popular temporal difference learning methods {\sf TD}$ (λ)$. Our main contribution is to provide a finite-analysis on the performance of this distributed {\sf TD}$(λ)$ algorithm for both constant and time-varying step sizes. The key idea in our analysis is to use the geometric mixing time $τ$ of the underlying Markov chain, that is, although the "noise" in our algorithm is Markovian, its dependence is very weak at samples spaced out at every $τ$. We provide an explicit upper bound on the convergence rate of the proposed method as a function of the network topology, the discount factor, the constant $λ$, and the mixing time $τ$. Our results also provide a mathematical explanation for observations that have appeared previously in the literature about the choice of $λ$. Our upper bound illustrates the trade-off between approximation accuracy and convergence speed implicit in the choice of $λ$. When $λ=1$, the solution will correspond to the best possible approximation of the value function, while choosing $λ= 0$ leads to faster convergence when the noise in the algorithm has large variance.

Zaiwei Chen, Sheng Zhang, Thinh T. Doan et al.

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.