Perspective Latents as an Architectural Condition for Causal Emergence in Active Inference Agents

A recent line of work measures causal emergence in reinforcement learning agents through Integrated Information Decomposition, reporting that $Φ_r$ grows with training and tracks reward improvement. For active inference, this raises the question of how reward-free predictive organization relates to such information-theoretic signatures. I test this within an active inference agent whose architecture separates a fast perception latent $z$ from a slow global latent $g$, where $g$ is driven by prediction error and structurally decoupled from policy gradients. In a reward-free environmental regime-switching protocol, $Φ_r$ concentrates in $g$; its aggregate magnitude is largely architectural and decreases with training. The substantive effect of learning becomes legible only at the atom-compositional level: decoupling flips sign from negative to positive and becomes regime-invariant under environmental change, while downward causation carries the regime-dependent adjustment. These results identify $g$ as the architectural locus of $Φ_r$-relevant temporal organization in an active inference agent, and argue against reading scalar $Φ_r$ as a direct index of learned integration.

Entropy Regularized Reinforcement Learning for Zero-Sum Stochastic Differential Games in a Regime-Switching Jump-Diffusion Process

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.

Deep numerical schemes for systems of Ergodic BSDEs with applications to regime-switching forward utilities

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.