Exponential families from a single KL identity
Exponential families encompass the distributions central to modern machine learning -- softmax, Gaussians, and Boltzmann distributions -- and underlie the theory of variational inference, entropy-regularized reinforcement learning, and RLHF. We isolate a simple identity for exponential families that expresses the KL difference $\mathrm{KL}(q \| p_{λ_2}) - \mathrm{KL}(q \| p_{λ_1})$ in terms of the log-partition function $A(λ)$ and the moment $μ_q$. Remarkably, this identity together with the single fact that $\mathrm{KL} \geq 0$ (with equality iff $p = q$) suffices, by direct substitution and rearrangement, to derive a cluster of results that are classically obtained by separate, heavier ...
| Authors | Marc Dymetman |
|---|---|
| Primary category | cs.LG |
| Published | 30 April 2026 |
| Last updated at source | 30 April 2026 |
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