Coherence Mechanisms for Provable Self-Improvement

Self-improvement is a critical capability for large language models and other intelligent systems, enabling them to refine their behavior and internal consistency without external supervision. Despite its importance, prior approaches largely rely on empirical heuristics and lack formal guarantees. In this paper, we propose a principled framework for self-improvement based on the concept of emph{coherence}, which requires that a model’s outputs remain consistent under task-preserving transformations of the input. We formalize this concept using projection-based mechanisms that update a baseline model to be coherent while remaining as close as possible to its original behavior. We provide rigorous theoretical guarantees that these mechanisms achieve emph{monotonic improvement}, measured by a reduction in expected Bregman divergence. Our analysis is comprehensive, covering both emph{direct} and emph{two-step} projection methods, and robustly extends these guarantees to non-realizable settings, empirical (finite-sample) distributions, and relaxed coherence constraints. Furthermore, we establish a general emph{characterization theorem}, showing that any mechanism with similar provable improvement guarantees must inherently conform to a coherence-based structure. This culminates in rigidity results under the demand for universal improvement, establishing coherence as a fundamental and, in a formal sense, necessary principle for provable self-improvement.