Offline evaluation of language models from usage logs is biased when model choice is confounded: the same user-side factors that influence which model is used can also influence how its output is judged, so raw comparisons of logged scores mix self-selected populations rather than estimating a common quantity of interest. A small randomized experiment can break this bias by overriding model choice, but in practice such experiments are scarce and costly. We study a three-source design that combines a large confounded observational log (OBS) for scale, a small randomized experiment (EXP) for unconfounded scoring, and an offline simulator (SIM) that replays candidate models on cached contexts. Our main result is an identification theorem showing that the randomized experiment and the simulator are together enough to recover causal model values; the observational log enters only afterward, to reduce estimation error rather than to make the causal comparison valid. Six estimator families are evaluated in a controlled semi-synthetic validation and in two real-task cached benchmarks for summarization and coding. No family dominates every regime; relative performance depends on the amount of unbiased EXP supervision and on how closely the target reward aligns with OBS-derived structure.

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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.

We consider tractable representations of probability distributions and the polytime operations they support. In particular, we consider a recently proposed arithmetic circuit representation, the Probabilistic Sentential Decision Diagram (PSDD). We show that PSDDs support a polytime multiplication operator, while they do not support a polytime operator for summing-out variables. A polytime multiplication operator makes PSDDs suitable for a broader class of applications compared to classes of arithmetic circuits that do not support multiplication. As one example, we show that PSDD multiplication leads to a very simple but effective compilation algorithm for probabilistic graphical models: represent each model factor as a PSDD, and then multiply them.

While the inverse sensitivity mechanism was shown to be instance optimal, it was only with respect to a class of unbiased mechanisms such that the most likely outcome matches the underlying data.

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Graph contrastivelearning (GCL), by training GNNs to maximize the correspondence between the representations of the same graph in its different augmented forms, may yield robust and transferable GNNs even without using labels.

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