Comparing two probability distributions is a basic building block of statistics and machine learning, and the right family is well understood: the Rényi divergences of order $α\in[0,\infty]$ are the unique family monotone under data processing and additive on independent products. Many problems instead compare more than two distributions at once -- multi-population fairness, multi-prior PAC-Bayes bounds, multi-hypothesis testing -- and the right multi-distribution generalization of the Rényi family has been an open question. We characterize it. Every functional of $W$-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of multi-way coincidence divergences $C_α(π_1,\dots,π_W) := -\log\int π_1^{α_1}\cdotsπ_W^{α_W}$ (with $\sum_k α_k = 1$) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones (the analogue of Rényi orders $>1$); a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary -- the destination of an explicit data-processing-monotone, product-additive divergence the others cannot reproduce -- and each is a clean limit of simplex-interior atoms. The same family arises from several independent routes -- the structural axioms, Kolmogorov-Nagumo means with Rényi's entropy axiomatics, classical entropy characterizations, multi-hypothesis testing error exponents, and a multi-lottery betting interpretation -- structural evidence that this is the canonical multi-distribution Rényi calculus rather than an artefact of any one axiomatic input. The two-prior case recovers the standard Rényi result; a worked $W=3$ instance, numerical verification, and a conditional extension round out the treatment.

Specifically,weconsider theRelational Network[21] architecture as a natural generalization of the DeepSets [32] architecture, and study their representational gap.

Persistent homology hasbecome animportant tool forextracting geometric and topological features from data, whose multi-scale features are summarized in a persistencediagram.

The standard definition of PAC learning (Valiant 1984) requires learners to succeed under all distributions -- even ones that are intractable to sample from. This stands in contrast to samplable PAC learning (Blum, Furst, Kearns, and Lipton 1993), where learners only have to succeed under samplable distributions. We study this distinction and show that samplable PAC substantially expands the power of efficient learners. We first construct a concept class that requires exponential sample complexity in standard PAC but is learnable with polynomial sample complexity in samplable PAC. We then lift this statistical separation to the computational setting and obtain a separation relative to a random oracle. Our proofs center around a new complexity primitive, explicit evasive sets, that we introduce and study. These are sets for which membership is easy to determine but are extremely hard to sample from. Our results extend to the online setting to similarly show how its landscape changes when the adversary is assumed to be efficient instead of computationally unbounded.

In [104] a powerful fully lifted (fl) probabilistic blirp interpolating mechanism was introduced. It arrived as a strong upgrade on partially lifted concepts from [100, 101] and the basic ones from [49, 84] (see also, e.g., [31, 32, 60, 106] for early considerations as well as [5, 64, 67, 101, 107] for a brief history, relevance, and development overview). While the range of applicability in a variety of scientific fields is rather wide, applications in random optimizations are of our prevalent interest. They became particularly fruitful over the last two decades (some of the most prominent examples include, compressed sensing, machine learning, and neural network statistical studies; see, e.g., [50, 72-75, 86-91, 108]). Characterizing typical behavior of their various features ranging from standard optimization metrics (objective values, optimal solutions, relations between optimizing variables) to associated algorithmic ones (accuracy, speed, convergence) became possible in large part due to a strong progress made in understanding and developing powerful comparison mechanisms. For example, many of the above performance metrics often exhibit the so-calledphase-transition (PT) phenomenon where they undergo an abrupt change as one moves from one region of system parameters to another.

This study investigates whether large language models, specifically GPT4, can match human capabilities in analogical reasoning within strategic decision making contexts. Using a novel experimental design involving source to target matching, we find that GPT4 achieves high recall by retrieving all plausible analogies but suffers from low precision, frequently applying incorrect analogies based on superficial similarities. In contrast, human participants exhibit high precision but low recall, selecting fewer analogies yet with stronger causal alignment. These findings advance theory by identifying matching, the evaluative phase of analogical reasoning, as a distinct step that requires accurate causal mapping beyond simple retrieval. While current LLMs are proficient in generating candidate analogies, humans maintain a comparative advantage in recognizing deep structural similarities across domains. Error analysis reveals that AI errors arise from surface level matching, whereas human errors stem from misinterpretations of causal structure. Taken together, the results suggest a productive division of labor in AI assisted organizational decision making where LLMs may serve as broad analogy generators, while humans act as critical evaluators, applying the most contextually appropriate analogies to strategic problems.