This analysis explains why conditional mutual information alone cannot certify escape and measures variation among intervention-conditioned updates rather than departure from the no-intervention law.
Abstract
Feedback-driven loops support iterative improvement in large language models, reinforcement learning, and autonomous discovery, yet their gains often diminish under repeated internal feedback. We study why closed-loop knowledge systems saturate and what external information can move them beyond their current attractors. We introduce a three-level operational framework in which knowledge states $x_t$ evolve through transition kernels $K_{\theta}$ indexed by a structural parameter $\theta$. The governing structure is defined as the observational equivalence class of $\theta$ induced by these kernels, while attractors and basins are properties of the fixed-$\theta$ dynamics. A structural intervention changes $\theta$ and produces a detectable kernel discrepancy on pre-specified probe states, making structural change falsifiable. Using a Lyapunov drift condition, we show that stable internal dynamics approach bounded stability regions with exponentially attenuated transients and a noise-controlled residual floor. We characterize escape through a metric condition on intervention-induced attractor displacement and a baseline-relative KL lower bound for increasing escape probability. This analysis also explains why conditional mutual information alone cannot certify escape: it measures variation among intervention-conditioned updates rather than departure from the no-intervention law. Case studies in LLM code repair, sparse-reward reinforcement learning, and Bayesian optimization use matched continuation controls to illustrate how feedback strength and alignment affect quality-improving escape. Our contribution is an operational connection among stability tools, measurable intervention effects, and cross-domain diagnostics.
A model-agnostic hybrid dynamics framework that blends a provably contracting nominal model with a flexible excursion model through an uncertainty-guided switching law is proposed, ensuring that each model operates within its reliability regime.
In performative reinforcement learning the deployed policy shapes the environment that generates the learner's future data, and the natural solution concept is a performatively stable policy that is optimal in the environment it induces. Existing convergence guarantees rely on Lipschitz sensitivity assumptions on the environment map $\pi \mapsto (P_\pi, r_\pi)$, which are hard to verify and fail in settings such as multi-agent best-response dynamics. We instead study stability for mixtures of policies, and show that the resulting picture is fundamentally different from performative prediction, where randomization removes the need for any sensitivity assumption. We distinguish local mixed stability, an occupancy-weighted first-order relaxation that we show is equivalent to stationarity, from global mixed stability, which certifies against arbitrary deviating policies. Our first result is that a weighted per-state Hedge dynamic drives the local stability gap to zero at an $O(1/\sqrt{T})$ rate for an arbitrary, possibly discontinuous, environment map, both with exact and with trajectory feedback. The two notions genuinely differ: we exhibit an instance where local stability is achieved exactly but every mixture has global stability gap bounded away from zero. For global stability we introduce a bounded transition range assumption, strictly weaker than Lipschitz sensitivity, under which unweighted per-state Hedge converges up to a floor of $O(\gamma\epsilon_P/(1-\gamma)^3)$, and we prove a matching-in-$\epsilon_P$ lower bound of $\Omega(\gamma\epsilon_P/(1-\gamma))$ under trajectory feedback, so this floor is unavoidable. Finally, we extend both notions to $n$-player performative Markov games, obtaining local stability with no assumption on the joint environment map or game structure, and global stability for performative Markov potential games.
Closed-loop human-AI systems generate high-dimensional behavioural trajectories whose collective dynamics remain obscure. Using 297,915 learners'adaptive-tutoring histories, we define semantic order variables before model fitting and test them in user-disjoint cohorts. The state exhibits reproducible basin-like flow and operationally defined, state-heterogeneous metastable-like kinetics. A construction-matched null distinguishes normalised-memory relaxation from a reproducible excess field. A four-term conditional mechanism recovers population drift (r = 0.946; learner-bootstrap 95% CI, 0.935-0.955). Predictive event-level self-supervised learning recovers the state and learned-plane flow; null-referenced corrections retain directional, partial-amplitude excess-field structure without full calibration. Shuffled-order training reverses learned-plane flow on ordered trajectories; support-alignment randomisation selectively reduces inward transport. Both axes remain linearly accessible without state supervision. Without cross-model fitting, the models share leading population drift (r = 0.866; learner-bootstrap 95% CI, 0.857-0.875) and persistence ordering; residual directions remain model-specific. These results identify an externally anchored leading-order effective field linking empirical dynamics, an interpretable mechanism and neural computation.
Min-Lin Wu, Xu Fang, Yi-Cheng Zhang et al.· 0 citations
How transient neural representations become integrated and stable enough to function as internal neural models remains incompletely understood. Grounded in efficient coding, Bayesian and predictive frameworks, recurrent and attractor dynamics, neural state-space models, and systems neuroscience, the Principle of Representation Integration for Stable Models (PRISM) proposes five operations: extraction, compression, integration, stabilization, and prediction/action. Here we developed PRISM-M, a minimal nine-equation recurrent dynamical realization with an explicit contraction condition (0 < J < 1), to examine whether these operations can generate persistent, context-sensitive, and prospectively informative model states. Seven simulation analyses showed persistent but revisable trajectories and a 0.209 context-dependent shift in the event-period model state. A 60 × 60 parameter sweep identified 719 rigid, 2,344 adaptive, and 537 high-gain trajectories within this structurally contractive parameter space, and the same regimes were recovered across 1,000 randomized environments. Perturbations of extraction, integration, and stabilization altered model trajectories, whereas the scalar compression perturbation had a small effect. The full PRISM state predicted the next model state more accurately than the model-state-only baseline (RMSE 0.0186 versus 0.0218), while performing similarly to an unconstrained ARX model. In the BART dataset, spatial fMRI states were distinguishable in 155 participants (69.7% accuracy; 33.3% chance), and inflation-related activity was modestly associated with pumping behavior in the 99 participants with matched behavioral data (β = 0.198, P = 0.040). PRISM-M provides a constrained, testable framework centered on four operational signatures of model-like organization structured representation, contextual integration, persistence or reconstructability, and prospective relevance. Author Summary The brain continually receives information from the outside world, the body, and its own ongoing activity, yet useful behavior requires more than simply detecting these signals. Neural information must be selected, organized, combined with context, maintained over time, and used to guide what happens next. We developed PRISM-M, a simple recurrent mathematical implementation of the Principle of Representation Integration for Stable Models (PRISM), to examine how these steps may work together within an explicit model. PRISM-M treats extraction, compression, integration, stabilization, and prediction/action as five interacting operations and tests whether they can produce internal states that persist over time while remaining able to change. Across seven simulations, the model generated context-sensitive and revisable states, remained mathematically stable across broad parameter ranges, and showed predictable effects when individual operations were altered. We also examined an independent human fMRI dataset from a sequential risk-taking task. The available data supported structured, condition-sensitive neural patterns and a modest relationship with behavior. They also indicated that temporally resolved recordings will be needed to test the full recurrent model directly. Together, these results provide a quantitative and testable framework for studying how neural representations may become stable internal neural models.
A central question in physical inference is whether strongly constrained dynamical systems can realize accurate input--output maps through their own finite-time evolution. We study this question in Kuramoto phase networks, whose deterministic dynamics form an input-conditioned gradient flow and whose predictions are read directly from output oscillators. As a constructive training approach, we develop a two-stage teacher--student procedure. A neural teacher is first converted into an explicit phase trajectory whose terminal oscillator activations reproduce the teacher outputs, and the Kuramoto parameters are trained by matching the student vector field along this prescribed path. Because accurate teacher-forced path matching does not ensure accurate autonomous inference, we then differentiate through the autonomous finite-time rollout and directly align its terminal output with the neural target. The resulting oscillator system, with $74$ oscillators, reaches mean test accuracies of $96.711\%$ on MNIST and $86.399\%$ on Fashion-MNIST. This capability persists across neural-teacher architectures, matched system sizes, thermal perturbations, and integration-grid refinement. Together, these results provide a constructive demonstration that a strongly constrained, small-sized Kuramoto gradient-flow system can be trained for high-accuracy finite-time inference through a direct oscillator readout.
This work provides a system-theoretic interpretation of generalization in learning-enabled dynamical systems arising in data-driven optimization and feedback control approximation, and establishes a matrix inequality-based certificate and a uniform stability bound that separates the one-sample sensitivity of the learned operator, and an algorithm-dependent dynamical gain.
Filippo Fabiani· 0 citations
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