We study robust Markov decision processes (RMDPs) with non-rectangular uncertainty sets, which capture interdependencies across states unlike traditional rectangular models. While non-rectangular robust policy evaluation is generally NP-hard, even in approximation, we identify a powerful class of $L_p$-bounded uncertainty sets that avoid these complexity barriers due to their structural simplicity. We further show that this class can be decomposed into infinitely many \texttt{sa}-rectangular $L_p$-bounded sets and leverage its structural properties to derive a novel dual formulation for $L_p$ RMDPs. This formulation provides key insights into the adversary's strategy and enables the development of the first robust policy evaluation algorithms for non-rectangular RMDPs. Empirical results demonstrate that our approach significantly outperforms brute-force methods, establishing a promising foundation for future investigation into non-rectangular robust MDPs.

In this paper, we establish the global convergence of the actor-critic algorithm with a significantly improved sample complexity of $O(\epsilon^{-3})$, advancing beyond the existing local convergence results. Previous works provide local convergence guarantees with a sample complexity of $O(\epsilon^{-2})$ for bounding the squared gradient of the return, which translates to a global sample complexity of $O(\epsilon^{-4})$ using the gradient domination lemma. In contrast to traditional methods that employ decreasing step sizes for both the actor and critic, we demonstrate that a constant step size for the critic is sufficient to ensure convergence in expectation. This key insight reveals that using a decreasing step size for the actor alone is sufficient to handle the noise for both the actor and critic. Our findings provide theoretical support for the practical success of many algorithms that rely on constant step sizes.

We present the first finite time global convergence analysis of policy gradient in the context of infinite horizon average reward Markov decision processes (MDPs). Specifically, we focus on ergodic tabular MDPs with finite state and action spaces. Our analysis shows that the policy gradient iterates converge to the optimal policy at a sublinear rate of $O\left({\frac{1}{T}}\right),$ which translates to $O\left({\log(T)}\right)$ regret, where $T$ represents the number of iterations. Prior work on performance bounds for discounted reward MDPs cannot be extended to average reward MDPs because the bounds grow proportional to the fifth power of the effective horizon. Thus, our primary contribution is in proving that the policy gradient algorithm converges for average-reward MDPs and in obtaining finite-time performance guarantees. In contrast to the existing discounted reward performance bounds, our performance bounds have an explicit dependence on constants that capture the complexity of the underlying MDP. Motivated by this observation, we reexamine and improve the existing performance bounds for discounted reward MDPs. We also present simulations to empirically evaluate the performance of average reward policy gradient algorithm.

Policy gradient methods have become a standard for training reinforcement learning agents in a scalable and efficient manner. However, they do not account for transition uncertainty, whereas learning robust policies can be computationally expensive. In this paper, we introduce robust policy gradient (RPG), a policy-based method that efficiently solves rectangular robust Markov decision processes (MDPs). We provide a closed-form expression for the worst occupation measure. Incidentally, we find that the worst kernel is a rank-one perturbation of the nominal. Combining the worst occupation measure with a robust Q-value estimation yields an explicit form of the robust gradient. Our resulting RPG can be estimated from data with the same time complexity as its non-robust equivalent. Hence, it relieves the computational burden of convex optimization problems required for training robust policies by current policy gradient approaches.

In robust Markov decision processes (RMDPs), it is assumed that the reward and the transition dynamics lie in a given uncertainty set. By targeting maximal return under the most adversarial model from that set, RMDPs address performance sensitivity to misspecified environments. Yet, to preserve computational tractability, the uncertainty set is traditionally independently structured for each state. This so-called rectangularity condition is solely motivated by computational concerns. As a result, it lacks a practical incentive and may lead to overly conservative behavior. In this work, we study coupled reward RMDPs where the transition kernel is fixed, but the reward function lies within an $\alpha$-radius from a nominal one. We draw a direct connection between this type of non-rectangular reward-RMDPs and applying policy visitation frequency regularization. We introduce a policy-gradient method, and prove its convergence. Numerical experiments illustrate the learned policy's robustness and its less conservative behavior when compared to rectangular uncertainty.

In Reinforcement Learning (RL), the goal of agents is to discover an optimal policy that maximizes the expected cumulative rewards. This objective may also be viewed as finding a policy that optimizes a linear function of its state-action occupancy measure, hereafter referred as Linear RL. However, many supervised and unsupervised RL problems are not covered in the Linear RL framework, such as apprenticeship learning, pure exploration and variational intrinsic control, where the objectives are non-linear functions of the occupancy measures. RL with non-linear utilities looks unwieldy, as methods like Bellman equation, value iteration, policy gradient, dynamic programming that had tremendous success in Linear RL, fail to trivially generalize. In this paper, we derive the policy gradient theorem for RL with general utilities. The policy gradient theorem proves to be a cornerstone in Linear RL due to its elegance and ease of implementability. Our policy gradient theorem for RL with general utilities shares the same elegance and ease of implementability. Based on the policy gradient theorem derived, we also present a simple sample-based algorithm. We believe our results will be of interest to the community and offer inspiration to future works in this generalized setting.

In reinforcement learning (RL), we are concerned with learning good policies for sequential decisionmaking problems modeled as Markov Decision Processes (MDPs) [29, 35]. MDPs assume that the transition model of the environment is fixed across training and testing, but this is often violated in practical applications. For example, when deploying a simulator-trained robot in reality, a notable challenge is the substantial disparity between the simulated environment and the intricate complexities of the real world, leading to potential subpar performance upon deployment. Such a mismatch may significantly degrade the performance of the trained policy in testing. To deal with this issue, the robust MDP (RMDP) framework has been introduced in [16, 24, 44], aiming to learn policies that are robust to perturbation of the transition model within an uncertainty set.

In Markov Decision Processes (MDPs), an agent interacts with the environment and learns to optimally behave in it [28]. However, the MDP solution may be very sensitive to little changes in the model parameters [23]. Hence we should be cautious applying the solution of the MDP, when the model is changing or when there is uncertainty in the model parameters. Robust MDPs provide a way to address this issue, where an agent can learn to optimally behave even when the model parameters are uncertain [15, 29, 18]. Another motivation to study robust MDPs is that they can lead to better generalization [33, 34, 25] compared to non-robust solutions.