Summary
This paper examines the confidence set of an estimator, defined by a likelihood ratio. The contributions stated within the work, alongside my corresponding queries, are outlined as follows:
* For generalized linear models, we theoretically analyze the geometry of the LR confidence sets under mild assumptions. We show their geometry is dictated by Bregman divergences of exponential families (Chowdhury et al., 2022).
*We show that the size of the confidence set is dictated by an online prediction game. The size of these sets depends on a sequence of estimators \{\theta_s\}_{s=1}^t that one uses to estimate the unknown parameter \theta_*. We discuss how to pick the estimator sequence in order to yield a provably small radius of the sets, by using the Follow-the-Regularized-Leader algorithm, which implements a regularized maximum-likelihood estimator. We prove that the radius of the confidence sets is nearly-worst-case optimal, and accordingly they yield nearly-worst-case regret bounds when used in generalized linear bandit applications. However, due to their data-dependent nature, they can be much tighter than this theory suggests.
*We analyze limitations of classical (un-weighted) LR sets when the underlying conditional observation model is not identifiable. In this case, the resulting (inevitable) estimation bias unnecessarily increases the size of the confidence sets. To mitigate this, we propose an adaptive reweighting scheme that decreases the effect of uninformed early bias of the estimator sequence on the size of the sets downstream. The reweighting does not affect the coverage guarantees of our sets, and utilizes an elegant connection to (robust) powered likelihoods (Wasserman et al., 2020).
* Thanks to the adaptive reweighting scheme, our sets are very practical as we showcase experimentally. We demonstrate that our method works well with exponential and non-exponential family likelihoods, and in parametric as well as in kernelized settings. We attribute their practical benefits to the fact that they do not depend on (possibly loose) worst-case parameters.
Strengths
The research question posed in this paper is notably intriguing and holds significant potential for various applications. Furthermore, the innovative approach adopted by the authors represents a substantial contribution to the field.
Weaknesses
While the paper offers several valuable insights, I noticed that the overall integration of its contributions could be improved. For instance, claimed contributions 1 and 3, which could potentially have been effectively merged, were instead treated separately. This compromise the contribution significantly from my perspective. Furthermore, I have concerns about the clarity and accuracy of some claims. For more specific observations, please refer to the "Questions" section. I believe addressing these issues would greatly enhance the coherence and validity of the work.
Questions
* The relationship between LR confidence set geometry and Bregman divergences is briefly discussed on page 6. It would be beneficial to have this connection elaborated upon more thoroughly in the main body of the text.
* Section 2 provides a clear explanation on how to select a sequence of $\{ \theta_s \}_{s=1}^t$ to minimize the radius of the confidence set. However, this connection becomes less clear in Section 3.
Moreover, the paper claims that the radius of the confidence set is nearly-worst-case optimal. Could you please provide further explanation as to why this is the case? While it's clear that the FTRL can lead to nearly-worst-case optimal regret for the online optimization problem, $\mathcal{R}_t$ defined in $(3)$ appears to be not identical, but smaller than the definition of regret in the online optimization problem. If $\mathcal{R}_t$ is significantly smaller than the regret of the FTRL algorithm, might our confidence set not be optimal?
* The introduction of a re-weighting scheme is intriguing. However, given that the re-weight $w_t$ is difficult to compute, could you provide any theoretical performance guarantees? Additionally, in Theorem 2, the bias estimate depends on \norm{\theta_*}_2^2. Could this potentially conflict with our objective to estimate the confidence set of \theta_*? Furthermore, in Section 3, we ascertain the geometry of the linear models assuming $w_t = 1$. Is it possible to perform a similar analysis incorporating the re-weighting scheme? Otherwise, the value of the re-weighting scheme might be discounted if we cannot incorporate it into the analysis of simple model
* A query arises from Theorem 3, where \theta \in \mathcal{C}_t satisfies a certain inequality. I am curious about the application of this inequality in estimating the confidence set of \theta_*, given that the Bregman distance involved depends on $\theta_*$, which is unknown. Is this result primarily of theoretical interest, or is it of practical use in this context?
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.