NeurIPS 2020

Minimax Bounds for Generalized Linear Models

Review 1

Summary and Contributions: Minimax risk bounds for Generalized Linear Models and Gaussian Linear models for L_2 and entropic risks. The results go beyond Lee-Courtade. using tools developed by Y. Wu. Gives interesting and improved results even when design matrix is not well structured (fat, rank deficient). Works for some random designs. The paper provides evidence that information theoretic methods can extract (sense?) dependencies on the structure of design matrix.

Strengths: Well-written. Timely, interesting and explainable results. Of interest to any theoretically inclined reader. I find the connexion between risks and ratio of Schatten 1-2 norms of design matrix particularly appealing.

Weaknesses: Nothing special

Correctness: Everything looks plausible. Could not spot any error.

Clarity: Very well written. Sober and well-chosen notation. Non-ambiguous definitions. Theorems are clearly stated, Discussion is sharp and readable.

Relation to Prior Work: Yes. Section 2.1. does a very good and useful job.

Reproducibility: Yes

Additional Feedback:

Review 2

Summary and Contributions: The paper considers generalized linear models and designs newer regret bounds which offer a significant improvement in the prior art. They also consider as special case \Gaussian linear models and when the linear coefficients are sampled from a Gaussian ensemble. I am not active in this area of research, nonetheless I found the results striking and engaging.

Strengths: The analysis is impressively novel and comprehensive. The paper is primarily a theoretical exposition to improve the regret bounds, which in and of itself is a worthy contribution. The ideas have been delineated logically and clearly.

Weaknesses: In general, paper is sound. I would be interested to see what happens if we relax the Gaussian ensemble assumption of 2.2, namely, what if there is mixture of Gaussian or a heavy-tailed sampling instead of Gaussian sampling? It would be also interesting to see some intuitive justification of the losses considered in equation (5) and (6) and behind Lemma 1. Some minor points, the notation of subscripts of \theta should be introduced before stating equation (5). Post-rebuttal : I am satisfied with the responses and would like to keep the score as is.

Correctness: Seems correct.

Clarity: Very well written.

Relation to Prior Work: Satisfactory.

Reproducibility: Yes

Additional Feedback:

Review 3

Summary and Contributions: This paper is devoted to establishing tight minimax lower bounds on the prediction error in generalized linear models. Crucially, the lower bounds established in this paper require weaker spectral properties of the design matrix, i.e. robust to both near-zero or extreme values in the spectrum. The main idea behind the proof is similar to [23], where the authors first reduce to a Bayesian entropic loss, and then apply the general relationship between the mutual information and Fisher information in [24] to lower bound the Bayesian entropic loss. The main contributions of this paper mainly include: 1. A general minimax lower bound on the prediction error for a large class of generalized linear models; 2. A minimax lower bound for sparse (Gaussian) estimation which requires weaker spectral properties of the design matrix; 3. A direct Bayesian lower bound instead of going through the multiple hypothesis testing and the metric entropies.

Strengths: The assumptions in this paper are general, claims are sound, and the results are interesting. It is particularly interesting to see an explicit Bayesian lower bound based on a natural Gaussian prior, whereas most prior work only showed even a weaker bound under a carefully constructed discrete prior. Also see the listed contributions above.

Weaknesses: 1. 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. 2. Specializing to sparse models, the contribution that weaker spectrum properties are now sufficient seems to be outweighed. First, some problems suffered in the previous approaches can be easily fixed. For example, the authors commented that when the design matrix M has two repeated columns, the previous lower bound becomes trivial. However, this can be easily fixed as follows. Assume wlog that the first two columns of M are the same, then we may simply fix \theta_1 = 0 and allow others to vary arbitrarily. In this way we effectively remove the first column of M, keep the same sparsity property, and only reduce the parameter dimension from d to d-1. Now applying the previous lower bound to the new and well-conditioned matrix gives the desired minimax rate. Also note that similar approaches can be taken even when half of the columns of M are repeated (and we reduce the dimension from d to d/2, which does not affect the rate analysis). Therefore, this comparison may seem slightly unfair and does not make a too strong case to me. Second, I do not fully understand why the authors treat the assumption k < n in previous work a ``crucial disadvantage". Note that for a typical constant noise level \sigma, the previous lower bound already shows that a sample complexity of n > k*log(ed/k) is necessary to achieve a constant statistical error. Also, for any \sigma, Eqn. (12) in this paper also shows that the case n > k gives a trivial error \sigma^2. This seems to suggest that one may wlog restrict to n > k for showing minimax lower bounds. 3. The tightness of the result is not sufficiently discussed. The authors claimed that their Theorem 3 is tight if either the largest or the smallest eigenvalues are of the same order. However, it seems that in those scenarios the previous lower bounds also give the tight answer. In other words, the authors did not explicitly construct an example such that the previous lower bound is not tight but the current bound becomes tight. Moreover, compared with the lower bound in (10), the authors did not show that the new bound provides a uniformly improvement over it. Finally, and most importantly, there is a missing logarithmic factor in the current lower bound, which is known to be necessary and important in sparse estimation. So missing the log factor seems to make the bound not very desirable in my opinion. Post rebuttal: The points #2, #3 are satisfactorily addressed in the rebuttal; please add these discussions to the final paper. However, my novelty concern over [23] is not adequately addressed. I took a closer look at both papers, and the only difference is on the upper bounds of the Fisher information, while other steps (bayes entropic loss, generalized van-trees inequality) are essentially the same. I agree that the current paper uses a different approach to upper bound the Fisher information: [23] used a Jensen's inequality (or a data-processing property of Fisher information) to relate the individual Fisher information to the trace of the entire Fisher information matrix (p.s. I do not understand why the authors call this a "single-letterization"); in the current paper, the individual Fisher information is studied directly by assuming a Gaussian prior and using the rotational invariance of the Gaussian distribution. However, I would prefer to treat this step as a direct and relatively straightforward computation of the Fisher information, and still do not think there is much technical innovation here. Given that this novelty concern remains, I decide to only increase my score from 5 to 6.

Correctness: The main proofs look correct to me.

Clarity: Yes.

Relation to Prior Work: The related literature was sufficiently discussed, but the authors should make more clarifications on the added novelty compared to [23].

Reproducibility: Yes

Additional Feedback: Some minor comments: Eqn. (11): the first \sup should be \inf. Sharpness: it might be better to clarify that the tight answer when the largest eigenvalues are of the same order is given by the first bound, and that when the smallest eigenvalues are of the same order is given by the second bound.

Review 4

Summary and Contributions: The authors establish a lower bound for the minimax prediction loss of generalized linear model through entropic loss.

Strengths: It improves on the current literature when the design matrix is ill-posed. They then discuss the situation for the Gaussian linear model and Gaussian design matrix.

Weaknesses: The contribution of the paper compared to its closely related work [19] seems to incremental. -- Update: the author's explanation partially addressed my concern.

Correctness: Probably. Did not check all details.

Clarity: Yes.

Relation to Prior Work: Based on my own reading, also as explained in the paper, the paper is very closed to reference [19]. The only improvement is that [19] considers estimating of entire $M\theta$, and here they consider \theta first and then estimate M\theta. The contribution of the paper seems to be limited.

Reproducibility: Yes

Additional Feedback: