cs.GL

This note provides a chronological account of Fr\'echet distances, starting with Maurice Fr\'echet's 1906 doctoral thesis on distances in abstract sets and tracing the Fr\'echet distance between polygonal curves and its algorithmic computation in the 1990s. It then continues with his 1957 paper on a coupling-based distance between probability laws with a brief glimpse of Wasserstein distance and optimal transport. We further attempt to draw connections between the distributional, coupling-based facet of Fr\'echet distances on probability laws and the geometric facet on curves. The note ends with a modern use case, the Fr\'echet Inception Distance (FID) in the era of deep generative model evaluation, interpretable as the Wasserstein-2 distance between multivariate Gaussians in a learned feature space. An appendix includes \TeX{}ified faithful English translations of Fr\'echet's 1906 thesis and 1957 paper, and L\'evy's 1950 note for reader convenience.

Dr. David Blackwell was a mathematician and statistician of the first rank, whose contributions to statistical theory, game theory, and decision theory predated many of the algorithmic breakthroughs that define modern artificial intelligence. This survey examines three of his most consequential theoretical results the Rao Blackwell theorem, the Blackwell Approachability theorem, and the Blackwell Informativeness theorem (comparison of experiments) and traces their direct influence on contemporary AI and machine learning. We show that these results, developed primarily in the 1940s and 1950s, remain technically live across modern subfields including Markov Chain Monte Carlo inference, autonomous mobile robot navigation (SLAM), generative model training, no-regret online learning, reinforcement learning from human feedback (RLHF), large language model alignment, and information design. NVIDIAs 2024 decision to name their flagship GPU architecture (Blackwell) provides vivid testament to his enduring relevance. We also document an emerging frontier: explicit Rao Blackwellized variance reduction in LLM RLHF pipelines, recently proposed but not yet standard practice. Together, Blackwell theorems form a unified framework addressing information compression, sequential decision making under uncertainty, and the comparison of information sources precisely the problems at the core of modern AI.