Towards Optimal Adversarial Robust Q-learning with Bellman Infinity-error (2402.02165v2)

Abstract: 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.

• The paper introduces the CAP framework to show that optimal policies retain consistency even under adversarial perturbations.
• It demonstrates that using the L∞-norm in Bellman error minimization significantly outperforms traditional L1-norm approaches.
• The CAR-DQN algorithm proves its robustness by delivering superior performance on Atari benchmarks in adversarial settings.

Analyzing Optimal Adversarial Robustness in Q-Learning via BeLLMan Infinity-Error

In this work, the authors explore the interplay between adversarial robustness and optimal policy derivation within deep reinforcement learning (DRL), particularly focusing on Q-learning methods. The paper addresses a significant challenge in DRL—developing policies robust to adversarial perturbations while maintaining optimal performance.

Key Contributions

The paper introduces three main contributions to the paper of adversarial robustness in DRL:

1. Consistency Assumption of Policy (CAP): The authors propose a theoretical framework, the Consistency Assumption of Policy, which hypothesizes the existence of intrinsic state neighborhoods where optimal actions remain consistent despite adversarial perturbations. CAP serves as a linchpin in proving the existence of deterministic and stationary Optimal Robust Policies (ORP), which align with BeLLMan optimal policies.
2. Necessity of $L^\infty$-norm: Through rigorous analysis, the paper identifies the substantial impact of using $L^{\infty}$-norm over other norms like $L^1$ in minimizing BeLLMan errors to attain adversarial robustness. The finding highlights why conventional DRL algorithms, based on $L^1$-norm, fall short in adversarial settings, thereby underscoring the need for $L^\infty$-norm to achieve optimal robust policies.
3. CAR-DQN Development: Building on their theoretical findings, the authors propose the Consistent Adversarial Robust Deep Q-Network (CAR-DQN), which leverages a surrogate objective to approximate BeLLMan infinity-error. This novel approach facilitates robust policy training against adversarial attacks, enhancing both natural and adversarial performance across diverse benchmarks.

Theoretical Insights

The paper’s theoretical exposition establishes CAP as a pivotal condition for achieving ORP. Empirical evidence supporting CAP indicates that most states adhere to this assumption, with exceptions arising in negligible amounts. Notably, under the CAP, the BeLLMan optimal policy is shown to be inherently robust, challenging the notion that robustness and optimality are inherently conflicting objectives.

Furthermore, the work provides a comprehensive stability analysis across varying Banach spaces, concluding that achieving robustness mandates attention to the $L^\infty$-norm in BeLLMan error minimizations.

Empirical Evaluation

Experimentation on classical Atari environments demonstrates CAR-DQN’s superior robustness compared to existing methods, such as those using Projected Gradient Descent (PGD) and convex relaxation techniques. CAR-DQN’s results affirm the alignment between theoretical predictions and practical performance improvements, particularly showcasing significant gains in environments like RoadRunner and BankHeist.

Practical Implications and Future Directions

CAR-DQN’s design offers a pathway to enhance the robustness of DRL agents without sacrificing performance under natural conditions. This robustness is critical for deploying DRL-enabled systems in real-world applications where adversarial conditions are likely. The algorithm's efficacy points to promising avenues for integrating $L^\infty$-oriented approaches in other RL paradigms, potentially extending to policy-based and continuous action settings.

The work opens questions around the generalizability of the consistency assumption beyond the tested environments, suggesting future research aimed at understanding CAP’s limits and exploring its applicability across diverse DRL frameworks.

In conclusion, this paper provides a significant step towards refining DRL methodologies to inherently support robust decision-making processes, advancing the field's capability to develop DRL applications resilient to adversarial interference.