Fast-rate bounds for Poisson regression

Establish a rigorous finite-sample excess prediction risk bound of order O(d/n) for Poisson regression under the fixed-design generalized linear model framework with the canonical link (ξ = x^Tθ) and negative log-likelihood loss, as defined in the paper’s prediction risk formulation. This aims to provide a fast-rate bound comparable to those available for linear and logistic regression within the same setting.

Background

Within the paper’s unified GLM risk analysis, the authors obtain non-asymptotic slow-rate bounds (on the order of √(d/n)) for several models and discuss when fast-rate results (O(d/n)) are achievable. For linear and logistic regression, certain conditions or closed-form analyses yield fast-rate bounds in finite samples or asymptotically.

However, for Poisson regression, despite presenting slow-rate guarantees and discussing related techniques, the authors note the absence of a rigorous finite-sample fast-rate bound and explicitly state that this remains open. The problem is framed in the fixed-design setting and uses the paper’s prediction risk definition, which separates training error from a stochastic term.

References

For Poisson regression, a rigorous finite-sample O(d/n) bound remains open, to the best of our knowledge.

— Basic Inequalities for First-Order Optimization with Applications to Statistical Risk Analysis  (2512.24999 - Paik et al., 31 Dec 2025) in Section “Risk analysis: ridge-regularized GLMs” — Comparison with existing literature

Despite our best efforts, we were unable to fully close this gap, and the $\varepsilon$ term remains. Establishing a final risk guarantee that is fully linear in $d$ is an interesting open problem, and solving it would provide a satisfactory closure to the study of Poisson regression under a Gaussian design.

— Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design  (2609.34757 - Lerasle et al., 28 Sep 2026) in Section 2, final paragraph of the discussion following Theorem \ref{th:mainth_riskbound_smalln}