arXiv subjects

Zaiwei Chen

Publications and source records attributed to Zaiwei Chen.

29 records · Page 2Linked to original sources

arXiv2023

A Finite-Sample Analysis of Payoff-Based Independent Learning in Zero-Sum Stochastic Games

We study two-player zero-sum stochastic games, and propose a form of independent learning dynamics called Doubly Smoothed Best-Response dynamics, which integrates a discrete and doubly smoothed variant of the best-response dynamics into temporal-difference (TD)-learning and minimax value iteration. The resulting dynamics are payoff-based, convergent, rational, and symmetric among players. Our main results provide finite-sample guarantees. In particular, we prove the first-known $\tilde{\mathcal{O}}(1/ε^2)$ sample complexity bound for payoff-based independent learning dynamics, up to a smoothing bias. In the special case where the stochastic game has only one state (i.e., matrix games), we provide a sharper $\tilde{\mathcal{O}}(1/ε)$ sample complexity. Our analysis uses a novel coupled Lyapunov drift approach to capture the evolution of multiple sets of coupled and stochastic iterates, which might be of independent interest.

cs.GT↗

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↗

arXiv2022

Global Convergence of Localized Policy Iteration in Networked Multi-Agent Reinforcement Learning

We study a multi-agent reinforcement learning (MARL) problem where the agents interact over a given network. The goal of the agents is to cooperatively maximize the average of their entropy-regularized long-term rewards. To overcome the curse of dimensionality and to reduce communication, we propose a Localized Policy Iteration (LPI) algorithm that provably learns a near-globally-optimal policy using only local information. In particular, we show that, despite restricting each agent's attention to only its $κ$-hop neighborhood, the agents are able to learn a policy with an optimality gap that decays polynomially in $κ$. In addition, we show the finite-sample convergence of LPI to the global optimal policy, which explicitly captures the trade-off between optimality and computational complexity in choosing $κ$. Numerical simulations demonstrate the effectiveness of LPI.

cs.LG↗

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

Nested Vehicle Routing Problem: Optimizing Drone-Truck Surveillance Operations

Unmanned aerial vehicles or drones are becoming increasingly popular due to their low cost and high mobility. In this paper we address the routing and coordination of a drone-truck pairing where the drone travels to multiple locations to perform specified observation tasks and rendezvous periodically with the truck to swap its batteries. We refer to this as the Nested-Vehicle Routing Problem (Nested-VRP) and develop a Mixed Integer Quadratically Constrained Programming (MIQCP) formulation with critical operational constraints, including drone battery capacity and synchronization of both vehicles during scheduled rendezvous. An enhancement of the MIQCP model for the Nested-VRP is achieved by deriving the equivalent Mixed Integer Linear Programming (MILP) formulation as well as leveraging lifting and Reformulation-Linearization techniques to strengthen the subtour elimination constraints of the drone. Given the NP-hard nature of the Nested-VRP, we further propose an efficient neighborhood search (NS) heuristic where we generate and improve on a good initial solution by iteratively solving the Nested-VRP on a local scale. We provide comparisons of both the exact approaches based on MIQCP or its enhanced formulations and NS heuristic methods with a relaxation lower bound in the cases of small and large problem sizes, and present the results of a computational study to show the effectiveness of the MIQCP model and its variants as well as the efficiency of the NS heuristic, including for a real-life instance with 631 locations. We envision that this framework will facilitate the planning and operations of combined drone-truck missions.

cs.DM↗

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↗

arXiv2021

Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization

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.

cs.LG↗

arXiv2021

Finite-Sample Analysis of Stochastic Approximation Using Smooth Convex Envelopes

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.

cs.LG↗

arXiv2021

Finite-Sample Analysis of Off-Policy TD-Learning via Generalized Bellman Operators

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.

cs.LG↗

arXiv2021

Finite-Sample Analysis of Off-Policy Natural Actor-Critic Algorithm

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.

cs.LG↗