A New Bound on the Cumulant Generating Function of Dirichlet Processes
<|reference_start|>A New Bound on the Cumulant Generating Function of Dirichlet Processes: In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) $X \sim \text{DP}(\alpha \nu_0)$, using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of $\alpha \mapsto \log \mathbb{E}_{X \sim \text{DP}(\alpha \nu_0)}[\exp( \mathbb{E}_X[\alpha f])]$, where $\mathbb{E}_X[f] = \int f dX$. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF $ \log\mathbb{E}_{X\sim \text{DP}(\alpha\nu_0)}{\exp(\mathbb{E}_{X}{[f]})} $ for any $\alpha > 0$. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence $\alpha\mathrm{KL}(\nu_0\Vert \cdot)$. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.<|reference_end|>
@article{perrault2024a,
    title={A New Bound on the Cumulant Generating Function of Dirichlet Processes},
    author={Pierre Perrault, Denis Belomestny, Pierre M'enard, 'Eric Moulines,
  Alexey Naumov, Daniil Tiapkin, Michal Valko},
    journal={arXiv preprint arXiv:2409.18621},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.18621},
    primaryClass={math.PR cs.IT math.IT}
}