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1. A Reinforcement Learning Look at Risk-Sensitive Linear Quadratic Gaussian Control

   Cui, Leilei ; Basar, Tamer ; Jiang, Zhong-Ping ( June 2023 , Proceedings of Machine Learning Research)

   Matni, N ; Morari, M ; Pappas, G J (Ed.)

   In this paper, we propose a robust reinforcement learning method for a class of linear discrete-time systems to handle model mismatches that may be induced by sim-to-real gap. Under the formulation of risk-sensitive linear quadratic Gaussian control, a dual-loop policy optimization algorithm is proposed to iteratively approximate the robust and optimal controller. The convergence and robustness of the dual-loop policy optimization algorithm are rigorously analyzed. It is shown that the dual-loop policy optimization algorithm uniformly converges to the optimal solution. In addition, by invoking the concept of small-disturbance input-to-state stability, it is guaranteed that the dual-loop policy optimization algorithm still converges to a neighborhood of the optimal solution when the algorithm is subject to a sufficiently small disturbance at each step. When the system matrices are unknown, a learning-based off-policy policy optimization algorithm is proposed for the same class of linear systems with additive Gaussian noise. The numerical simulation is implemented to demonstrate the efficacy of the proposed algorithm. 

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2. Robust Reinforcement Learning for Risk-Sensitive Linear Quadratic Gaussian Control

   https://doi.org/10.1109/TAC.2024.3397928  

   Cui, Leilei ; Başar, Tamer ; Jiang, Zhong-Ping ( January 2024 , IEEE Transactions on Automatic Control)

   This paper proposes a novel robust reinforcement learning framework for discrete-time linear systems with model mismatch that may arise from the sim-to-real gap. A key strategy is to invoke advanced techniques from control theory. Using the formulation of the classical risk-sensitive linear quadratic Gaussian control, a dual-loop policy optimization algorithm is proposed to generate a robust optimal controller. The dual-loop policy optimization algorithm is shown to be globally and uniformly convergent, and robust against disturbances during the learning process. This robustness property is called small-disturbance input-to-state stability and guarantees that the proposed policy optimization algorithm converges to a small neighborhood of the optimal controller as long as the disturbance at each learning step is relatively small. In addition, when the system dynamics is unknown, a novel model-free off-policy policy optimization algorithm is proposed. Finally, numerical examples are provided to illustrate the proposed algorithm. 

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3. A Lyapunov characterization of robust policy optimization

   https://doi.org/10.1007/s11768-023-00163-w  

   Cui, Leilei ; Jiang, Zhong-Ping ( August 2023 , Control Theory and Technology)

   In this paper, we study the robustness property of policy optimization (particularly Gauss–Newton gradient descent algorithm which is equivalent to the policy iteration in reinforcement learning) subject to noise at each iteration. By invoking the concept of input-to-state stability and utilizing Lyapunov’s direct method, it is shown that, if the noise is sufficiently small, the policy iteration algorithm converges to a small neighborhood of the optimal solution even in the presence of noise at each iteration. Explicit expressions of the upperbound on the noise and the size of the neighborhood to which the policies ultimately converge are provided. Based on Willems’ fundamental lemma, a learning-based policy iteration algorithm is proposed. The persistent excitation condition can be readily guaranteed by checking the rank of the Hankel matrix related to an exploration signal. The robustness of the learning-based policy iteration to measurement noise and unknown system disturbances is theoretically demonstrated by the input-to-state stability of the policy iteration. Several numerical simulations are conducted to demonstrate the efficacy of the proposed method. 

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4. Distributionally Robust Q-Learning

   Liu, Zijian ; Bai, Qinxun ; Blanchet, Jose H. ; Dong, Perry ; Xu, Wei ; Zhou, Zhengqing ; Zhou, Zhengyuan ( July 2022 , Proceedings of Machine Learning Research)

   Chaudhuri, Kamalika ; Jegelka, Stefanie ; Song, Le ; Szepesvari, Csaba ; Niu, Gang ; Sabato, Sivan (Ed.)

   Reinforcement learning (RL) has demonstrated remarkable achievements in simulated environments. However, carrying this success to real environments requires the important attribute of robustness, which the existing RL algorithms often lack as they assume that the future deployment environment is the same as the training environment (i.e. simulator) in which the policy is learned. This assumption often does not hold due to the discrepancy between the simulator and the real environment and, as a result, and hence renders the learned policy fragile when deployed. In this paper, we propose a novel distributionally robust Q-learning algorithm that learns the best policy in the worst distributional perturbation of the environment. Our algorithm first transforms the infinite-dimensional learning problem (since the environment MDP perturbation lies in an infinite-dimensional space) into a finite-dimensional dual problem and subsequently uses a multi-level Monte-Carlo scheme to approximate the dual value using samples from the simulator. Despite the complexity, we show that the resulting distributionally robust Q-learning algorithm asymptotically converges to optimal worst-case policy, thus making it robust to future environment changes. Simulation results further demonstrate its strong empirical robustness. 

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5. Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization

   https://doi.org/10.1287/opre.2021.2151  

   Cen, Shicong ; Cheng, Chen ; Chen, Yuxin ; Wei, Yuting ; Chi, Yuejie ( January 2022 , Operations Research)

   Natural policy gradient (NPG) methods are among the most widely used policy optimization algorithms in contemporary reinforcement learning. This class of methods is often applied in conjunction with entropy regularization—an algorithmic scheme that encourages exploration—and is closely related to soft policy iteration and trust region policy optimization. Despite the empirical success, the theoretical underpinnings for NPG methods remain limited even for the tabular setting. This paper develops nonasymptotic convergence guarantees for entropy-regularized NPG methods under softmax parameterization, focusing on discounted Markov decision processes (MDPs). Assuming access to exact policy evaluation, we demonstrate that the algorithm converges linearly—even quadratically, once it enters a local region around the optimal policy—when computing optimal value functions of the regularized MDP. Moreover, the algorithm is provably stable vis-à-vis inexactness of policy evaluation. Our convergence results accommodate a wide range of learning rates and shed light upon the role of entropy regularization in enabling fast convergence. 

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