In almost all these applications the quality of the embedding is measured by variousaverage case criteria.

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, minimalistic -- weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-ofthe-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest.

In such a setting, distribution-free risk bounds are impossible, and the best one could hope for is a fully empirical data-dependent bound.

In such a setting, distribution-free risk bounds are impossible, and the best one could hope for is a fully empirical data-dependent bound.

This behavior usually stems from the over-smoothing phenomenon.

The challenge of finding compromises between agent proposals is fundamental to AI sub-fields such as argumentation, mediation, and negotiation. Building on this tradition, Elkind et al. (2021) introduced a process for coalition formation that seeks majority-supported proposals preferable to the status quo, using a metric space where each agent has an ideal point. The crucial step in this iterative process involves identifying compromise proposals around which agent coalitions can unite. How to effectively find such compromise proposals, however, remains an open question. We address this gap by formalizing a holistic model that encompasses agent bounded rationality and uncertainty and developing AI models to generate such compromise proposals. We focus on the domain of collaboratively writing text documents -- e.g., to enable the democratic creation of a community constitution. We apply NLP (Natural Language Processing) techniques and utilize LLMs (Large Language Models) to create a semantic metric space for text and develop algorithms to suggest suitable compromise points. To evaluate the effectiveness of our algorithms, we simulate various coalition formation processes and demonstrate the potential of AI to facilitate large-scale democratic text editing, such as collaboratively drafting a constitution, an area where traditional tools are limited.