NeurIPS 2020

### Review 1

Summary and Contributions: This paper studies best arm identification in linear bandits in the fixed confidence setting. The authors provide an asymptotically optimal algorithm and evaluate it through experiments. They also analyze a best arm identification problem on the unit sphere.

Strengths: The proposed algorithm is simple and the analysis in the paper is rigorous. It does seem relevant to the pure exploration bandits community to have a track-and-stop algorithm for linear bandits.

Weaknesses: The authors claim that their experiments suggest that their algorithm has a strong empirical performance. I am unconvinced because they only run experiments on one set up. In order to support this claim, they need to consider a variety of experimental setups. One of the central claims in this paper is they provide a practical, asymptotically optimal algorithm, so it is problematic that they do not provide more support demonstrating its practicability. It does not seem very difficult to come up with a practical, yet asymptotically optimal algorithm (up to constants). One can combine the RAGE algorithm with the explore-and-then-verify framework of [13] (see "Nearly Optimal Sampling Algorithms for Combinatorial Pure Exploration" by Chen et al. for an example of this sort of algorithm in the setting of combinatorial bandits). It seems like the contribution here is to get rid of a constant and to come up with an algorithm in the style of [18]. Regarding their analysis of \epsilon-good arm identification for the sphere, it seems that one could just construct an (\epsilon/2)-net of the sphere and apply an algorithm for finite number of bandits to this. Is there anything fundamentally more difficult about this problem setting? It is unclear how often to update the allocation in practice. If this confers computational advantages in practice, it is important to provide practitioners with guidance on this.

Correctness: Yes.

Clarity: The paper is clear.

Relation to Prior Work: Yes.

Reproducibility: Yes

Additional Feedback: It would be useful if the authors clarified what made solving this problem challenging.

### Review 2

Summary and Contributions: The paper considers the best arm identification problem in linear bandits. It proposes the track-and-stop type algorithm and proves it achieves theoretical lower bound. The empirical results seem promising as well.

Strengths: It proposes a novel algorithm for BAI in linear bandits, which is an important problem. The result of achieving theoretical lower bound is significant. As far as I checked, the proof seems correct.

Weaknesses: To me, there is no explicit weakness.

Correctness: I could not find any mistake in the proof

Clarity: Yes

Relation to Prior Work: Yes

Reproducibility: Yes

### Review 3

Summary and Contributions: This work gives a quite clean and asymptotically optimal answer for best arm identification in linear bandits. It builds upon the track-and-stop framework of Garivier and Kaufmann, with the interesting twist that the authors consider here a lazy version (which does not hurt the asymptotic optimality).

Strengths: Best arm identification in linear bandits has potentially many applications, and the proposed algorithm has both a solid theoretical guarantee and seems to perform well in practice.

Weaknesses: The problem is not exactly fresh", with already many papers on the same exact topic. The same can be said about the proposed technique, which by now is getting quite classical. In some ways it is a bit frustrating to see an asymptotic analysis, when arguably the field went through a resurgence when it was realized that finite-time guarantees actually make a difference (I am referring to Auer et al. versus Lai and Robbins type analyses).

Correctness: The paper appears to be correct.

Clarity: The paper is well written, and I especially appreciated the very lucid presentation in Section 3.

Relation to Prior Work: The prior work is appropriate.

Reproducibility: Yes

### Review 4

Summary and Contributions: I thank the author for the rebuttal. I still think this paper could be valuable for the community. However after discussing with the other reviewers and the meta reviewer we would have liked a discussion on the validity (or not) of the Franke-Wolfe algorithm to update the allocation (see the recent paper by Degenne for instance) and a better discussion and more details on the complexity not depending on the number of arms. This paper introduces a new strategy for the problem of best arm identification with fixed confidence in stochastic linear bandits. This strategy allows to achieve the known lower bound asymptotically almost surely and in expectation (main results of the paper). The strategy is based on the track and stop principle: it tracks the optimal distribution over the arms using the estimated regression parameter. This optimal proportion can be updated as often as one wants. The authors also provide a first result for best arm identification in linear bandits for a continuous set of arms compared to the usual finite set of arms. Finally n experiment compares the introduced algorithm Lazy TS with the state-of-the-art RAGE algorithm.

Strengths: The approach devised by the authors is novel compared to existing work. Most existing work relies on rounding procedures which implies additional computational costs and does not reach the lower bound when the confidence level delta tends to 0 (e.g. XY-adaptive allocation from Soare et al, RAGE from Fiez et al. and Tao et al. (does not reach lower bound but does not use a rounding procedure)). A very recent paper, "Gamification of Pure Exploration for Linear Bandits" by Degenne et al., presented at ICML 2020. https://proceedings.icml.cc/static/paper_files/icml/2020/4512-Supplemental.pdf also obtains an asymptotically optimal algorithm for fixed-confidence pure exploration in linear bandits but the strategy derived therein is different from the one of this paper. The authors might want to cite this paper. The introduced algorithm, Lazy TS, seems to perform well in practice on the experiment considered in the paper

Weaknesses: More classical experiments usually considered for best arm identification in linear bandits could be done in order to better assess the performance of the algorithm in practice compared to existing algorithms (see eg the experiments in the RAGE paper). How bad can be the c_{A_0} constant?

Correctness: The claims look correct but I did not check the proofs in the supplementary material.

Clarity: Overall the paper is well written. I had sometimes a hard time connecting the different results of section 3.3. which describe how the sampling rule is designed. Some terms are not very clear to me such as "certainty equivalence principle".

Relation to Prior Work: The related work and the advantages of the proposed strategy is clearly discussed. As mentioned above only one very recent paper deriving the same result but with a different strategy is missing.

Reproducibility: Yes

Additional Feedback: Could the authors describe the idea of the track and stop principle when mentioning it in the introduction as this is one of the most important tools of the strategy derived in the paper? This would make the paper better self contained. Usually in best arm identification we care about the performance of the estimator hat mu in the directions (a - a') and we use an upper bound of <hat mu - mu, a -a'> as we do not need to be good in all the directions to discriminate arms and find the best one. These directions appear in the stopping rule and in the lower bound but not in the bound use for the least square estimator. Can the authors detail why they do not need such a bound compared to existing work? Can you confirm that the goal of A_0 is to prevent the minimum eigenvalue of the design matrix to be too small (which could be arbitrarily small depending on the set of arms and the optimal distributions)? However it seems to me that c_{A_0} could still be very small in practice: for instance in R^2, if the set of arms is made of 2 arms with same norm and the angle between these 2 arms tends to 0. I would like an interpretation of the stopping rule. For the continuous set of arms why does one need to restrict the study to the unit sphere? Why does the continuous set make the analysis challenging as written l 243? - l 130: do we really need it for the paper without the supplementary materials? - l 141: L appears twice: L = max ... <= L - a is both used to denote a member of the set of arms and an integer in {1, ..., K}. - A_0 is not specified in Algorithm 1. In the experiments it is chosen at random but is it really the best strategy? Shouldn't one pick the best A_0 if possible? - I would use V instead of A in Algorithm 1 to denote the design matrix. - The algorithm called Lazy Track and Stop is referred to as LTS at some places and as Lazy TS in other places. - l 249: be a subset, that forms - l 176 : Franke Broader impact: please also state that recommender systems can be used with bad intention such as influence people opinion during elections. A lot of people have been raising issues of such systems lately.