arXiv subjects

Thinh T. Doan

Publications and source records attributed to Thinh T. Doan.

At least 19 recordsLinked to original sources

arXiv2026

Nonasymptotic CLT and Error Bounds for Linear Two-Time-Scale Stochastic Approximation

We consider linear two-time-scale stochastic approximation algorithms driven by martingale noise. Recent applications in machine learning motivate the need to understand finite-time error rates, but conventional stochastic approximation analyses focus on either asymptotic convergence in distribution or finite-time bounds that are far from optimal. Prior work on asymptotic central limit theorems (CLTs) suggests that two-time-scale algorithms may be able to achieve $1/\sqrt{K}$ error in expectation, with a constant given by the expected norm of the limiting Gaussian vector. However, the best known finite-time rates are much slower. We derive the first nonasymptotic Wasserstein-1 CLT for linear two-time-scale stochastic approximation with Polyak-Ruppert averaging driven by martingale difference noise. As a corollary, we show that the expected error achieved by Polyak-Ruppert averaging decays at rate $1/\sqrt{K}$, which significantly improves on the rates of convergence in prior works.

cs.LG↗

arXiv2026

CBF-based Probabilistic Safe Navigation under Unknown Nonlinear Obstacle Dynamics

Safe navigation for an ego vehicle in uncertain environments characterized by dynamic obstacles with unknown nonlinear dynamics is a challenging problem of significant practical interest. Existing approaches in the literature either lack formal safety guarantees, require full model knowledge, or fail to account for the risk associated with the vehicle's exact body geometry and the temporal evolution of uncertainty between sampling instants. In this paper, we propose a data-driven observer for the unknown obstacle dynamics that generates an alpha-confidence set flow, which is exactly transformed into a Control Barrier Function (CBF) to enforce (1-alpha)-probability safety. The proposed framework accommodates nonlinear ego vehicle dynamics of arbitrary relative degree, as demonstrated through case studies involving first- and second-order dynamics of an unmanned surface vehicle.

eess.SY↗

arXiv2026

Internal State-Based Policy Gradient Methods for Partially Observable Markov Potential Games

This letter studies multi-agent reinforcement learning in partially observable Markov potential games. Solving this problem is challenging due to partial observability, decentralized information, and the curse of dimensionality. First, to address the first two challenges, we leverage the common information framework, which allows agents to act based on both shared and local information. Second, to ensure tractability, we study an internal state that compresses accumulated information, preventing it from growing unboundedly over time. We then implement an internal state-based natural policy gradient method to find Nash equilibria of the Markov potential game. Our main contribution is to establish a non-asymptotic convergence bound for this method. Our theoretical bound decomposes into two interpretable components: a statistical error term that also arises in standard Markov potential games, and an approximation error capturing the use of finite-state controllers. Finally, simulations across multiple partially observable environments demonstrate that the proposed method using finite-state controllers achieves consistent improvements in performance compared to the setting where only the current observation is used.

cs.MA↗

arXiv2026

Finite-Time Analysis of Projected Two-Time-Scale Stochastic Approximation

We study the finite-time convergence of projected linear two-time-scale stochastic approximation with constant step sizes and Polyak--Ruppert averaging. We establish an explicit mean-square error bound, decomposing it into two interpretable components, an approximation error determined by the constrained subspace and a statistical error decaying at a sublinear rate, with constants expressed through restricted stability margins and a coupling invertibility condition. These constants cleanly separate the effect of subspace choice (approximation errors) from the effect of the averaging horizon (statistical errors). We illustrate our theoretical results through a number of numerical experiments on both synthetic and reinforcement learning problems.

eess.SY↗

arXiv2026

Fast Two-Time-Scale Stochastic Gradient Method with Applications in Reinforcement Learning

Two-time-scale optimization is a framework introduced in Zeng et al. (2024) that abstracts a range of policy evaluation and policy optimization problems in reinforcement learning (RL). Akin to bi-level optimization under a particular type of stochastic oracle, the two-time-scale optimization framework has an upper level objective whose gradient evaluation depends on the solution of a lower level problem, which is to find the root of a strongly monotone operator. In this work, we propose a new method for solving two-time-scale optimization that achieves significantly faster convergence than the prior arts. The key idea of our approach is to leverage an averaging step to improve the estimates of the operators in both lower and upper levels before using them to update the decision variables. These additional averaging steps eliminate the direct coupling between the main variables, enabling the accelerated performance of our algorithm. We characterize the finite-time convergence rates of the proposed algorithm under various conditions of the underlying objective function, including strong convexity, Polyak-Lojasiewicz condition, and general non-convexity. These rates significantly improve over the best-known complexity of the standard two-time-scale stochastic approximation algorithm. When applied to RL, we show how the proposed algorithm specializes to novel online sample-based methods that surpass or match the performance of the existing state of the art. Finally, we support our theoretical results with numerical simulations in RL.

math.OC↗

arXiv2025

Accelerating Multi-Task Temporal Difference Learning under Low-Rank Representation

We study policy evaluation problems in multi-task reinforcement learning (RL) under a low-rank representation setting. In this setting, we are given $N$ learning tasks where the corresponding value function of these tasks lie in an $r$-dimensional subspace, with $r<N$. One can apply the classic temporal-difference (TD) learning method for solving these problems where this method learns the value function of each task independently. In this paper, we are interested in understanding whether one can exploit the low-rank structure of the multi-task setting to accelerate the performance of TD learning. To answer this question, we propose a new variant of TD learning method, where we integrate the so-called truncated singular value decomposition step into the update of TD learning. This additional step will enable TD learning to exploit the dominant directions due to the low rank structure to update the iterates, therefore, improving its performance. Our empirical results show that the proposed method significantly outperforms the classic TD learning, where the performance gap increases as the rank $r$ decreases. From the theoretical point of view, introducing the truncated singular value decomposition step into TD learning might cause an instability on the updates. We provide a theoretical result showing that the instability does not happen. Specifically, we prove that the proposed method converges at a rate $\mathcal{O}(\frac{\ln(t)}{t})$, where $t$ is the number of iterations. This rate matches that of the standard TD learning.

cs.LG↗

arXiv2024

Finite-Time Complexity of Online Primal-Dual Natural Actor-Critic Algorithm for Constrained Markov Decision Processes

We consider a discounted cost constrained Markov decision process (CMDP) policy optimization problem, in which an agent seeks to maximize a discounted cumulative reward subject to a number of constraints on discounted cumulative utilities. To solve this constrained optimization program, we study an online actor-critic variant of a classic primal-dual method where the gradients of both the primal and dual functions are estimated using samples from a single trajectory generated by the underlying time-varying Markov processes. This online primal-dual natural actor-critic algorithm maintains and iteratively updates three variables: a dual variable (or Lagrangian multiplier), a primal variable (or actor), and a critic variable used to estimate the gradients of both primal and dual variables. These variables are updated simultaneously but on different time scales (using different step sizes) and they are all intertwined with each other. Our main contribution is to derive a finite-time analysis for the convergence of this algorithm to the global optimum of a CMDP problem. Specifically, we show that with a proper choice of step sizes the optimality gap and constraint violation converge to zero in expectation at a rate $\mathcal{O}(1/K^{1/6})$, where K is the number of iterations. To our knowledge, this paper is the first to study the finite-time complexity of an online primal-dual actor-critic method for solving a CMDP problem. We also validate the effectiveness of this algorithm through numerical simulations.

math.OC↗

arXiv2024

Accelerated Multi-Time-Scale Stochastic Approximation: Optimal Complexity and Applications in Reinforcement Learning and Multi-Agent Games

Multi-time-scale stochastic approximation is an iterative algorithm for finding the fixed point of a set of $N$ coupled operators given their noisy samples. It has been observed that due to the coupling between the decision variables and noisy samples of the operators, the performance of this method decays as $N$ increases. In this work, we develop a new accelerated variant of multi-time-scale stochastic approximation, which significantly improves the convergence rates of its standard counterpart. Our key idea is to introduce auxiliary variables to dynamically estimate the operators from their samples, which are then used to update the decision variables. These auxiliary variables help not only to control the variance of the operator estimates but also to decouple the sampling noise and the decision variables. This allows us to select more aggressive step sizes to achieve an optimal convergence rate. Specifically, under a strong monotonicity condition, we show that for any value of $N$ the $t^{\text{th}}$ iterate of the proposed algorithm converges to the desired solution at a rate $\widetilde{O}(1/t)$ when the operator samples are generated from a single from Markov process trajectory. A second contribution of this work is to demonstrate that the objective of a range of problems in reinforcement learning and multi-agent games can be expressed as a system of fixed-point equations. As such, the proposed approach can be used to design new learning algorithms for solving these problems. We illustrate this observation with numerical simulations in a multi-agent game and show the advantage of the proposed method over the standard multi-time-scale stochastic approximation algorithm.

math.OC↗

arXiv2024

Resilient Two-Time-Scale Local Stochastic Gradient Descent for Byzantine Federated Learning

We study local stochastic gradient descent methods for solving federated optimization over a network of agents communicating indirectly through a centralized coordinator. We are interested in the Byzantine setting where there is a subset of $f$ malicious agents that could observe the entire network and send arbitrary values to the coordinator to disrupt the performance of other non-faulty agents. The objective of the non-faulty agents is to collaboratively compute the optimizer of their respective local functions under the presence of Byzantine agents. In this setting, prior works show that the local stochastic gradient descent method can only return an approximate of the desired solutions due to the impacts of Byzantine agents. Whether this method can find an exact solution remains an open question. In this paper, we will address this open question by proposing a new variant of the local stochastic gradient descent method. Under similar conditions that are considered in the existing works, we will show that the proposed method converges exactly to the desired solutions. We will provide theoretical results to characterize the convergence properties of our method, in particular, the proposed method convergences at an optimal rate $\mathcal{O}(1/k)$ in both strongly convex and non-convex settings, where $k$ is the number of iterations. Finally, we will present a number of simulations to illustrate our theoretical results.

math.OC↗

arXiv2024

A Two-Time-Scale Stochastic Optimization Framework with Applications in Control and Reinforcement Learning

We study a new two-time-scale stochastic gradient method for solving optimization problems, where the gradients are computed with the aid of an auxiliary variable under samples generated by time-varying MDPs controlled by the underlying optimization variable. These time-varying samples make gradient directions in our update biased and dependent, which can potentially lead to the divergence of the iterates. In our two-time-scale approach, one scale is to estimate the true gradient from these samples, which is then used to update the estimate of the optimal solution. While these two iterates are implemented simultaneously, the former is updated "faster" than the latter. Our first contribution is to characterize the finite-time complexity of the proposed two-time-scale stochastic gradient method. In particular, we provide explicit formulas for the convergence rates of this method under different structural assumptions, namely, strong convexity, PL condition, and general non-convexity. We apply our framework to various policy optimization problems. First, we look at the infinite-horizon average-reward MDP with finite state and action spaces and derive a convergence rate of $O(k^{-2/5})$ for the online actor-critic algorithm under function approximation, which recovers the best known rate derived specifically for this problem. Second, we study the linear-quadratic regulator and show that an online actor-critic method converges with rate $O(k^{-2/3})$. Third, we use the actor-critic algorithm to solve the policy optimization problem in an entropy regularized Markov decision process, where we also establish a convergence of $O(k^{-2/3})$. The results we derive for both the second and third problem are novel and previously unknown in the literature. Finally, we briefly present the application of our framework to gradient-based policy evaluation algorithms in reinforcement learning.

math.OC↗

arXiv2024

Bayesian meta learning for trustworthy uncertainty quantification

We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive distributions with a pre-specified probability. Furthermore, we propose, Trust-Bayes, a novel optimization framework for Bayesian meta learning which is cognizant of trustworthy uncertainty quantification without explicit assumptions on the prior model/distribution of the functions. We characterize the lower bounds of the probabilities of the ground truth being captured by the specified intervals and analyze the sample complexity with respect to the feasible probability for trustworthy uncertainty quantification. Monte Carlo simulation of a case study using Gaussian process regression is conducted for verification and comparison with the Meta-prior algorithm.

stat.ML↗

arXiv2024

Natural Policy Gradient and Actor Critic Methods for Constrained Multi-Task Reinforcement Learning

Multi-task reinforcement learning (RL) aims to find a single policy that effectively solves multiple tasks at the same time. This paper presents a constrained formulation for multi-task RL where the goal is to maximize the average performance of the policy across tasks subject to bounds on the performance in each task. We consider solving this problem both in the centralized setting, where information for all tasks is accessible to a single server, and in the decentralized setting, where a network of agents, each given one task and observing local information, cooperate to find the solution of the globally constrained objective using local communication. We first propose a primal-dual algorithm that provably converges to the globally optimal solution of this constrained formulation under exact gradient evaluations. When the gradient is unknown, we further develop a sampled-based actor-critic algorithm that finds the optimal policy using online samples of state, action, and reward. Finally, we study the extension of the algorithm to the linear function approximation setting.

math.OC↗

arXiv2024

Fast Nonlinear Two-Time-Scale Stochastic Approximation: Achieving $O(1/k)$ Finite-Sample Complexity

This paper proposes to develop a new variant of the two-time-scale stochastic approximation to find the roots of two coupled nonlinear operators, assuming only noisy samples of these operators can be observed. Our key idea is to leverage the classic Ruppert-Polyak averaging technique to dynamically estimate the operators through their samples. The estimated values of these averaging steps will then be used in the two-time-scale stochastic approximation updates to find the desired solution. Our main theoretical result is to show that under the strongly monotone condition of the underlying nonlinear operators the mean-squared errors of the iterates generated by the proposed method converge to zero at an optimal rate $O(1/k)$, where $k$ is the number of iterations. Our result significantly improves the existing result of two-time-scale stochastic approximation, where the best known finite-time convergence rate is $O(1/k^{2/3})$. We illustrate this result by applying the proposed method to develop new reinforcement learning algorithms with improved performance.

math.OC↗

arXiv2023

Resilient Federated Learning under Byzantine Attack in Distributed Nonconvex Optimization with 2-f Redundancy

We study the problem of Byzantine fault tolerance in a distributed optimization setting, where there is a group of $N$ agents communicating with a trusted centralized coordinator. Among these agents, there is a subset of $f$ agents that may not follow a prescribed algorithm and may share arbitrarily incorrect information with the coordinator. The goal is to find the optimizer of the aggregate cost functions of the honest agents. We will be interested in studying the local gradient descent method, also known as federated learning, to solve this problem. However, this method often returns an approximate value of the underlying optimal solution in the Byzantine setting. Recent work showed that by incorporating the so-called comparative elimination (CE) filter at the coordinator, one can provably mitigate the detrimental impact of Byzantine agents and precisely compute the true optimizer in the convex setting. The focus of the present work is to provide theoretical results to show the convergence of local gradient methods with the CE filter in a nonconvex setting. We will also provide a number of numerical simulations to support our theoretical results.

math.OC↗

arXiv2023

Connected Superlevel Set in (Deep) Reinforcement Learning and its Application to Minimax Theorems

The aim of this paper is to improve the understanding of the optimization landscape for policy optimization problems in reinforcement learning. Specifically, we show that the superlevel set of the objective function with respect to the policy parameter is always a connected set both in the tabular setting and under policies represented by a class of neural networks. In addition, we show that the optimization objective as a function of the policy parameter and reward satisfies a stronger "equiconnectedness" property. To our best knowledge, these are novel and previously unknown discoveries. We present an application of the connectedness of these superlevel sets to the derivation of minimax theorems for robust reinforcement learning. We show that any minimax optimization program which is convex on one side and is equiconnected on the other side observes the minimax equality (i.e. has a Nash equilibrium). We find that this exact structure is exhibited by an interesting robust reinforcement learning problem under an adversarial reward attack, and the validity of its minimax equality immediately follows. This is the first time such a result is established in the literature.

cs.LG↗

arXiv2023

Distributed Dual Subgradient Methods with Averaging and Applications to Grid Optimization

We study finite-time performance of a recently proposed distributed dual subgradient (DDSG) method for convex constrained multi-agent optimization problems. The algorithm enjoys performance guarantees on the last primal iterate, as opposed to those derived for ergodic means for vanilla DDSG algorithms. Our work improves the recently published convergence rate of $\Ocal(\log T/\sqrt{T})$ with decaying step-sizes to $\Ocal(1/\sqrt{T})$ with constant step-size on a metric that combines suboptimality and constraint violation. We then numerically evaluate the algorithm on three grid optimization problems. Namely, these are tie-line scheduling in multi-area power systems, coordination of distributed energy resources in radial distribution networks, and joint dispatch of transmission and distribution assets. The DDSG algorithm applies to each problem with various relaxations and linearizations of the power flow equations. The numerical experiments illustrate various properties of the DDSG algorithm--comparison with vanilla DDSG, impact of the number of agents, and why Nesterov-style acceleration fails in DDSG settings.

math.OC↗

arXiv2022

Regularized Gradient Descent Ascent for Two-Player Zero-Sum Markov Games

We study the problem of finding the Nash equilibrium in a two-player zero-sum Markov game. Due to its formulation as a minimax optimization program, a natural approach to solve the problem is to perform gradient descent/ascent with respect to each player in an alternating fashion. However, due to the non-convexity/non-concavity of the underlying objective function, theoretical understandings of this method are limited. In our paper, we consider solving an entropy-regularized variant of the Markov game. The regularization introduces structure into the optimization landscape that make the solutions more identifiable and allow the problem to be solved more efficiently. Our main contribution is to show that under proper choices of the regularization parameter, the gradient descent ascent algorithm converges to the Nash equilibrium of the original unregularized problem. We explicitly characterize the finite-time performance of the last iterate of our algorithm, which vastly improves over the existing convergence bound of the gradient descent ascent algorithm without regularization. Finally, we complement the analysis with numerical simulations that illustrate the accelerated convergence of the algorithm.

math.OC↗

arXiv2022

Convergence Rates of Distributed Consensus over Cluster Networks: A Two-Time-Scale Approach

We study the popular distributed consensus method over networks composed of a number of densely connected clusters with a sparse connection between them. In these cluster networks, the method often constitutes two-time-scale dynamics, where the internal nodes within each cluster reach consensus quickly relative to the aggregate nodes across clusters. Our main contribution is to provide the rate of the distributed consensus method, which characterize explicitly the impacts of the internal and external graphs on the performance of this method. Our main result shows that this rate converges exponentially and only scales with a few number of nodes, which is relatively small to the size of the network. The key technique in our analysis is to consider a Lyapunov function which captures the impacts of different time-scale dynamics on the convergence of the method. Our approach avoids using model reduction, which is the typical way according to singular perturbation theory and relies on relatively simple definitions of the slow and fast variables. In addition, Lyapunov analysis allows us to derive the rate of distributed consensus methods over cluster networks, which is missing from the existing works using singular perturbation theory. We illustrate our theoretical results by a number of numerical simulations over different cluster networks.

math.OC↗