Entropy-Regularized Reinforcement Learning for Linear-Quadratic Stackelberg Differential Games in Regime-Switching Diffusion Models
Authors: Congde Hu, Danping Li, Lin Xu, Wenying Xu
Abstract
Stackelberg differential games (SDGs) provide a powerful framework for hierarchical decision-making in stochastic and continuous-time environments, yet their solution remains computationally challenging due to the complexity of traditional dynamic programming and Hamilton-Jacobi-Bellman-Isaacs (HJBI) methods, especially in high-dimensional systems. This paper proposes an entropy-regularized reinforcement learning (ERRL) approach for linear-quadratic SDGs (LQ-SDGs) within a continuous-time diffusion framework governed by Markovian regime switching. The key innovation lies in deriving exploratory weakly-coupled HJBI equations with entropy regularization, which promotes stochastic policies that actively avoid suboptimal equilibria -- a limitation of classical SDG methods. Neural networks are integrated to approximate regime-dependent value functions and solve high-dimensional partial differential equations (PDEs) efficiently, while a novel sampling technique enhances computational tractability. Numerical results demonstrate the effectiveness of the framework compared to conventional approaches, particularly in escaping suboptimal traps through exploratory policies. The study highlights the critical role of entropy regularization and neural network approximations in achieving robust solutions for hierarchical decision-making problems under abrupt environmental shifts.
To address parameter misspecification and sudden structural environmental changes in conventional stochastic differential game (SDG) frameworks, this paper introduces a distributional control approach that characterizes optimal strategies as probability distributions over actions, conditioned on the continuous state, the discrete regime state, and parameters. This forms a reinforcement learning framework for entropy-regularized zero-sum stochastic differential games (ERRL-ZSSDGs) in a regime-switching jump-diffusion process. Using the dynamic programming principle (DPP), we derive the associated coupled systems of Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, from which equilibrium strategies are expressed via gradients of the value function. For linear-quadratic problems, semi-analytical solutions for both value function and equilibrium strategies are obtained by solving a system of coupled ordinary differential equations (ODEs). In more general settings, an Actor-Critic policy improvement algorithm is developed to approximate the value functions and equilibrium policies across different regimes. The method is applied to an investment game, and numerical examples illustrate the effect of the temperature parameter and regime transitions on optimal policies and values.
Entropy regularization is widely used in continuous-time reinforcement learning (RL) to reduce sensitivity to environmental perturbations, yet its robustness benefits lack a rigorous theoretical foundation. This paper establishes the first robustness guarantees for entropy-regularized continuous-time Markov decision processes. We show that maximizing an entropy-regularized objective yields a lower bound on a worst-case robust RL problem with joint reward and transition perturbations. We analytically characterize the induced robust sets and prove that they expand monotonically with the regularization strength, justifying the empirical observation that stronger entropy improves robustness. In contrast to prior discrete-time analyses, our results remove the intractable state-distribution entropy term and provide guarantees invariant to action frequency. Experiments on queueing network control and market making confirm our theory, showing that entropy-regularized policies outperform greedy and ε-greedy baselines under dynamics perturbations.
Neural networks applied to sequential decision-making tasks typically rely on latent representations of environment states. While environment dynamics dictate how semantic states evolve, the corresponding latent transitions are usually left implicit, creating a potential misalignment between the two. We propose to model latent dynamics explicitly by drawing an analogy between Markov decision process (MDP) trajectories and ordinary differential equation (ODE) flows: in both cases, the current state fully determines its successors. Building on this view, we introduce a neural ODE-based regularization method that enforces latent embeddings to follow consistent ODE flows, thereby aligning representation learning with environment dynamics. Although broadly applicable to deep learning agents, we demonstrate its effectiveness in reinforcement learning by integrating it into Actor-Critic algorithms. Our approach yields major performance gains across various standard Atari benchmarks for A2C and gridworld environments for PPO.