We establish a new class of minimax prediction error bounds for generalized linear models. Our bounds significantly improve previous results when the design matrix is poorly structured, including natural cases where the matrix is wide or does not have full column rank. Apart from the typical $L_2$ risks, we study a class of entropic risks which recovers the usual $L_2$ prediction and estimation risks, and demonstrate that a tight analysis of Fisher information can uncover underlying structural dependency in terms of the spectrum of the design matrix. The minimax approach we take differs from the traditional metric entropy approach, and can be applied to many other settings.

The novelty of this paper seems questionable, mainly in view of [23]. Specifically, [23] studied a similar problem for generalized linear models where the only difference seems to be that the estimation error was considered instead of the prediction error. The technical steps are also very close to each other: both work reduced to Bayesian entropic loss, then the result of [24] was invoked to show that an upper bound on the Fisher information is sufficient, and finally the authors provided upper bounds on the Fisher information. Of course the last step is different; however this difference does not seem to add too much novelty. First, some problems suffered in the previous approaches can be easily fixed.

The reviewers enjoyed reading your paper, and have made several useful comments for the camera-ready version. It would be particularly useful for readers if the authors were clear on the precise technical differences between their work and the work of Lee-Courtade, which they follow closely.

We establish a new class of minimax prediction error bounds for generalized linear models. Our bounds significantly improve previous results when the design matrix is poorly structured, including natural cases where the matrix is wide or does not have full column rank. Apart from the typical L_2 risks, we study a class of entropic risks which recovers the usual L_2 prediction and estimation risks, and demonstrate that a tight analysis of Fisher information can uncover underlying structural dependency in terms of the spectrum of the design matrix. The minimax approach we take differs from the traditional metric entropy approach, and can be applied to many other settings.