Sharp Concentration for Gibbs Distributions under Gaussian Score Noise

Gibbs distributions, or softmax distributions under a uniform reference measure, convert scores into probabilistic decisions. We study their stability when the scores are perturbed by correlated Gaussian estimation errors. The forward Kullback-Leibler (KL) divergence between the population and noisy Gibbs distributions is exactly a nonlinear log-partition remainder after its first-order response to the score error is removed. Our main result gives a sharp, dimension-independent concentration bound for this remainder: after normalization by the largest variance of a pairwise score difference, its centered positive moment-generating function is bounded by that of a Gamma random variable with shape $1/2$. The result holds for arbitrary finite action sets, reference probabilities, temperatures, Gaussian correlations, and noise magnitudes, and a balanced two-action problem approaches equality in the small-noise limit. The proof combines decreasing covariance envelopes under Gaussian smoothing with an explicit bound on the nonzero average gradient caused by asymmetric priors. We derive high-probability KL certificates, $O(1/ϵ)$ sample-complexity bounds, a data-dependent guarantee for unknown covariance, stronger results for structured action spaces, and trajectory-level guarantees for deterministic entropy-regularized planning. Numerical experiments examine tightness and the practical size of these certificates.