Ensuring the robustness of deep reinforcement learning (DRL) agents against adversarial attacks is critical for their trustworthy deployment. Recent research highlights the challenges of achieving state-adversarial robustness and suggests that an optimal robust policy (ORP) does not always exist, complicating the enforcement of strict robustness constraints. In this paper, we further explore the concept of ORP. We first introduce the Intrinsic State-adversarial Markov Decision Process (ISA-MDP), a novel formulation where adversaries cannot fundamentally alter the intrinsic nature of state observations. ISA-MDP, supported by empirical and theoretical evidence, universally characterizes decision-making under state-adversarial paradigms. We rigorously prove that within ISA-MDP, a deterministic and stationary ORP exists, aligning with the Bellman optimal policy. Our findings theoretically reveal that improving DRL robustness does not necessarily compromise performance in natural environments. Furthermore, we demonstrate the necessity of infinity measurement error (IME) in both $Q$-function and probability spaces to achieve ORP, unveiling vulnerabilities of previous DRL algorithms that rely on $1$-measurement errors. Motivated by these insights, we develop the Consistent Adversarial Robust Reinforcement Learning (CAR-RL) framework, which optimizes surrogates of IME. We apply CAR-RL to both value-based and policy-based DRL algorithms, achieving superior performance and validating our theoretical analysis.

Establishing robust policies is essential to counter attacks or disturbances affecting deep reinforcement learning (DRL) agents. Recent studies explore state-adversarial robustness and suggest the potential lack of an optimal robust policy (ORP), posing challenges in setting strict robustness constraints. This work further investigates ORP: At first, we introduce a consistency assumption of policy (CAP) stating that optimal actions in the Markov decision process remain consistent with minor perturbations, supported by empirical and theoretical evidence. Building upon CAP, we crucially prove the existence of a deterministic and stationary ORP that aligns with the Bellman optimal policy. Furthermore, we illustrate the necessity of $L^{\infty}$-norm when minimizing Bellman error to attain ORP. This finding clarifies the vulnerability of prior DRL algorithms that target the Bellman optimal policy with $L^{1}$-norm and motivates us to train a Consistent Adversarial Robust Deep Q-Network (CAR-DQN) by minimizing a surrogate of Bellman Infinity-error. The top-tier performance of CAR-DQN across various benchmarks validates its practical effectiveness and reinforces the soundness of our theoretical analysis.