We study learning under regime variation, where the learner, its memory state, and the evaluative conditions may evolve over time. This paper is a foundational and structural contribution: its goal is to define the core learning-theoretic objects required for such settings and to establish their first theorem-supporting consequences. The paper develops a regime-varying framework centered on admissible transport, protected-core preservation, and evaluator-aware learning evolution. It records the immediate closure consequences of admissibility, develops a structural obstruction argument for faithful fixed-ontology reduction in genuinely multi-regime settings, and introduces a protected-stability template together with explicit numerical and symbolic witnesses on controlled subclasses, including convex and deductive settings. It also establishes theorem-layer results on evaluator factorization, morphisms, composition, and partial kernel-level alignment across semantically commensurable layers. A worked two-regime example makes the admissibility certificate, protected evaluative core, and regime-variation cost explicit on a controlled subclass. The symbolic component is deliberately restricted in scope: the paper establishes a first kernel-level compatibility result together with a controlled monotonic deductive witness. The manuscript should therefore be read as introducing a structured learning-theoretic framework for regime-varying learning together with its first theorem-supporting layer, not as a complete quantitative theory of all learning systems.

It is well known that Empirical Risk Minimization (ERM) may attain minimax suboptimal rates in terms of the mean squared error (Birgé and Massart, 1993). In this paper, we prove that, under relatively mild assumptions, the suboptimality of ERM must be due to its bias. Namely, the variance error term of ERM (in terms of the bias and variance decomposition) enjoys the minimax rate. In the fixed design setting, we provide an elementary proof of this result using the probabilistic method. Then, we extend our proof to the random design setting for various models. In addition, we provide a simple proof of Chatterjee's admissibility theorem (Chatterjee, 2014, Theorem 1.4), which states that in the fixed design setting, ERM cannot be ruled out as an optimal method, and then we extend this result to the random design setting. We also show that our estimates imply stability of ERM, complementing the main result of Caponnetto and Rakhlin (2006) for non-Donsker classes. Finally, we highlight the somewhat irregular nature of the loss landscape of ERM in the non-Donsker regime, by showing that functions can be close to ERM, in terms of $L_2$ distance, while still being far from almost-minimizers of the empirical loss.

Flow matching (FM) has shown promising results in data-driven planning. However, it inherently lacks formal guarantees for ensuring state and action constraints, whose satisfaction is a fundamental and crucial requirement for the safety and admissibility of planned trajectories on various systems. Moreover, existing FM planners do not ensure the dynamical consistency, which potentially renders trajectories inexecutable. We address these shortcomings by proposing SAD-Flower, a novel framework for generating Safe, Admissible, and Dynamically consistent trajectories. Our approach relies on an augmentation of the flow with a virtual control input. Thereby, principled guidance can be derived using techniques from nonlinear control theory, providing formal guarantees for state constraints, action constraints, and dynamic consistency. Crucially, SAD-Flower operates without retraining, enabling test-time satisfaction of unseen constraints. Through extensive experiments across several tasks, we demonstrate that SAD-Flower outperforms various generative-model-based baselines in ensuring constraint satisfaction.

In this research note, we show the relationship between two non-admissible argumentation framework semantics: cogent and weakly admissible semantics. We prove that, while cogent extensions are weakly admissible, the converse is not true.

Heuristic functions are central to the performance of search algorithms such as A-star, where admissibility - the property of never overestimating the true shortest-path cost - guarantees solution optimality. Recent deep learning approaches often disregard admissibility and provide limited guarantees on generalization beyond the training data. This paper addresses both of these limitations. First, we pose heuristic learning as a constrained optimization problem and introduce Cross-Entropy Admissibility (CEA), a loss function that enforces admissibility during training. On the Rubik's Cube domain, this method yields near-admissible heuristics with significantly stronger guidance than compressed pattern database (PDB) heuristics. Theoretically, we study the sample complexity of learning heuristics. By leveraging PDB abstractions and the structural properties of graphs such as the Rubik's Cube, we tighten the bound on the number of training samples needed for A-star to generalize. Replacing a general hypothesis class with a ReLU neural network gives bounds that depend primarily on the network's width and depth, rather than on graph size. Using the same network, we also provide the first generalization guarantees for goal-dependent heuristics.

We present a theoretical framework for an Exam Readiness Index (ERI): a composite, blueprint-aware score R in [0,100] that summarizes a learner's readiness for a high-stakes exam while remaining interpretable and actionable. The ERI aggregates six signals -- Mastery (M), Coverage (C), Retention (R), Pace (P), Volatility (V), and Endurance (E) -- each derived from a stream of practice and mock-test interactions. We formalize axioms for component maps and the composite, prove monotonicity, Lipschitz stability, and bounded drift under blueprint re-weighting, and show existence and uniqueness of the optimal linear composite under convex design constraints. We further characterize confidence bands via blueprint-weighted concentration and prove compatibility with prerequisite-admissible curricula (knowledge spaces / learning spaces). The paper focuses on theory; empirical study is left to future work.

In this work, we broaden the investigation of admissibility notions in the context of assumption-based argumentation (ABA). More specifically, we study two prominent alternatives to the standard notion of admissibility from abstract argumentation, namely strong and weak admissibility, and introduce the respective preferred, complete and grounded semantics for general (sometimes called non-flat) ABA. To do so, we use abstract bipolar set-based argumentation frameworks (BSAFs) as formal playground since they concisely capture the relations between assumptions and are expressive enough to represent general non-flat ABA frameworks, as recently shown. While weak admissibility has been recently investigated for a restricted fragment of ABA in which assumptions cannot be derived (flat ABA), strong admissibility has not been investigated for ABA so far. We introduce strong admissibility for ABA and investigate desirable properties. We furthermore extend the recent investigations of weak admissibility in the flat ABA fragment to the non-flat case. We show that the central modularization property is maintained under classical, strong, and weak admissibility. We also show that strong and weakly admissible semantics in non-flat ABA share some of the shortcomings of standard admissible semantics and discuss ways to address these.

This article develops a novel framework for modal logic based on the idea of stratified actualization, rather than the classical model of global possible worlds. Traditional Kripke semantics treat modal operators as quantification over fully determinate alternatives, neglecting the local, dynamic, and often asymmetric nature of actualization processes. We propose a system Stratified Actualization Logic (SAL) in which modalities are indexed by levels of ontological stability, interpreted as admissibility regimes. Each modality operates over a structured layer of possibility, grounded in the internal coherence of transitions between layers. We formally define the syntax and semantics of SAL, introduce its axioms, and prove soundness and completeness. Applications are discussed in connection with temporal becoming, quantum decoherence domains, and modal metaphysics. The result is a logic that captures the ontological structure of actualization without recourse to abstract possible worlds, offering a stratified alternative to standard modal realism.

Diffusion models excel at creating images and videos thanks to their multimodal generative capabilities. These same capabilities have made diffusion models increasingly popular in robotics research, where they are used for generating robot motion. However, the stochastic nature of diffusion models is fundamentally at odds with the precise dynamical equations describing the feasible motion of robots. Hence, generating dynamically admissible robot trajectories is a challenge for diffusion models. To alleviate this issue, we introduce DDAT: Diffusion policies for Dynamically Admissible Trajectories to generate provably admissible trajectories of black-box robotic systems using diffusion models. A sequence of states is a dynamically admissible trajectory if each state of the sequence belongs to the reachable set of its predecessor by the robot's equations of motion. To generate such trajectories, our diffusion policies project their predictions onto a dynamically admissible manifold during both training and inference to align the objective of the denoiser neural network with the dynamical admissibility constraint. The auto-regressive nature of these projections along with the black-box nature of robot dynamics render these projections immensely challenging. We thus enforce admissibility by iteratively sampling a polytopic under-approximation of the reachable set of a state onto which we project its predicted successor, before iterating this process with the projected successor. By producing accurate trajectories, this projection eliminates the need for diffusion models to continually replan, enabling one-shot long-horizon trajectory planning. We demonstrate that our framework generates higher quality dynamically admissible robot trajectories through extensive simulations on a quadcopter and various MuJoCo environments, along with real-world experiments on a Unitree GO1 and GO2.

It is well known that Empirical Risk Minimization (ERM) may attain minimax suboptimal rates in terms of the mean squared error (Birgé and Massart, 1993). In this paper, we prove that, under relatively mild assumptions, the suboptimality of ERM must be due to its bias. Namely, the variance error term of ERM (in terms of the bias and variance decomposition) enjoys the minimax rate. In the fixed design setting, we provide an elementary proof of this result using the probabilistic method. Then, we extend our proof to the random design setting for various models.