Exponential families from a single KL identity

Problem
This preprint addresses a gap in the theoretical understanding of exponential families, which are foundational to various machine learning paradigms, including variational inference and reinforcement learning. The paper presents a novel identity that simplifies the derivation of several classical results related to Kullback-Leibler (KL) divergence, which have traditionally required more complex arguments. By isolating a single KL identity, the author aims to streamline the theoretical framework surrounding exponential families and their applications.

Method
The core technical contribution is the derivation of a KL divergence identity expressed as $\mathrm{KL}(q | p_{λ_2}) - \mathrm{KL}(q | p_{λ_1}) = A(λ_2) - A(λ_1) + μ_q(λ_2 - λ_1)$, where $A(λ)$ is the log-partition function and $μ_q$ is the moment associated with distribution $q$. This identity allows for the derivation of various results, including a generalized three-point identity for arbitrary reference distributions, Pythagorean theorems for I-projections, and the convexity of the log-partition function. The author employs direct substitution and rearrangement techniques to derive these results, which are typically obtained through more intricate methods. The paper also recovers the gradient formula for the log-partition function and the Bregman representation of KL divergence within the same framework.

Results
The paper does not present empirical results or benchmark comparisons, as it primarily focuses on theoretical derivations. However, the implications of the derived identities are significant for understanding the structure of exponential families and their applications in machine learning. The author claims that the results derived from the single KL identity can lead to more efficient algorithms in variational inference and reinforcement learning contexts, although specific performance metrics or comparisons to existing methods are not provided.

Limitations
The author acknowledges that the work is primarily theoretical and does not include empirical validation of the derived results. Additionally, the paper does not explore the practical implications of the derived identities in real-world applications or provide a comprehensive comparison with existing theoretical frameworks. An obvious limitation is the lack of experimental results that would demonstrate the utility of the theoretical contributions in practical scenarios.

Why it matters
This work has significant implications for downstream research in machine learning, particularly in areas that rely on exponential families, such as variational inference and reinforcement learning. By simplifying the theoretical underpinnings of KL divergence and its relationship with exponential families, the paper may facilitate the development of more efficient algorithms and models. The derived identities could lead to advancements in entropy-regularized control and reinforcement learning from human feedback (RLHF), potentially improving the performance and interpretability of models in these domains.

Authors: Marc Dymetman
Source: arXiv:2604.28036
URL: https://arxiv.org/abs/2604.28036v1