NeurIPS 2020

Statistical Efficiency of Thompson Sampling for Combinatorial Semi-Bandits

Review 1

Summary and Contributions: This paper studied the stochastic combinatorial multi-armed bandit (CMAB) problem under two families of distributions: mutually independent outcomes and multivariate sub-Gaussian outcomes. It improved the regret bound of the previous work on Combinatorial Thompson Sampling (CTS) for CMAB under bounded mutually independent outcomes. It proposed a new CTS algorithm with Gaussian priors for a more general sub-Gaussian family and proved a non-trivial upper bound of the algorithm. Experiments demonstrate the effectiveness of the proposed algorithm.

Strengths: It provided a new analysis of CTS under bounded mutually independent outcomes that improve the regret bound from the previous work. For a more general sub-Gaussian family of distribution, it proposed a new algorithm with Gaussian priors and rigorously proved the regret upper bound, which is the main technical contribution. It discussed the insights and the novelty of this regret analysis.

Weaknesses: ---After Rebuttal--- I read the authors' rebuttal and would like to maintain my score of 7. ----- The motivation of considering multivariate sub-Gaussian family of distribution is missing. It is better to explain more or show some concrete applications. The analysis for bounded mutually independent outcomes is still based on the analysis of Wang and Chen [2018], though it modified some lemmas in the proof.

Correctness: This paper is technically sound. Most of the claims are well supported by theoretical analysis.

Clarity: The paper is clearly written and well organized. I suggest double-checking the reference page since some papers are not cited correctly, for example, Agrawal and Goyal [2012], Degenne and Perchet [2016], Wang and Chen [2018] are all published in conferences, not only in arxiv.

Relation to Prior Work: It has detailed comparison with previous works.

Reproducibility: Yes

Additional Feedback:

Review 2

Summary and Contributions: This paper studies the different problem instances of the stochastic combinatorial multi-armed bandits with semi-bandit feedback (CMAB). The considered instances have different dependence structures on arm distributions. The authors propose computationally efficient policies using Combinatorial Thompson Sampling (CTS) policy with Beta priors (CTS-BETA) and Gaussian priors (CTS-GAUSSIAN) for above-mentioned problem instances. They show that the (theoretical and empirical) performance of their policies is better than state-of-art policies.

Strengths: The paper's major contribution is the computationally efficient policy (CTS-GAUSSIAN) for the CMAB problem instance where the joint arm distribution is C-sub-Gaussian, where C is a semi-definite matrix. The authors have shown that the regret upper bound of CTS-GAUSSIAN is the same as state-of-art computationally inefficient ESCB policy. The authors propose CLIP CTS-GAUSSIAN policy for a special case of CMAB problem instance where the joint arm distribution is C-sub-Gaussian with $\lambda \in R^n_+$. The CLIP CTS-GAUSSIAN policy has the same or better regret bounds than the CTS-GAUSSIAN policy. The authors also improve the existing results for the CMAB problem instance where arm distributions are mutually independent.

Weaknesses: CTS-GAUSSIAN policy needs two input parameters: vector D and \beta. It is not clear how to choose vector D if arm distributions are unknown. Also, it is not mentioned that how to select the parameter \beta in the CTS-GAUSSIAN policy.

Correctness: I haven't found any flaws in the paper. I also skimmed the proofs, and it seems sound to me.

Clarity: Overall, the paper is well-written and is easy to follow.

Relation to Prior Work: Authors clearly distinguish between their work from the existing literature.

Reproducibility: Yes

Additional Feedback: Minor comment: 1. Mention the value of \beta used in the experiments. ************************************************************************************* Post Rebuttal: I have read the rebuttal and comments of other reviewers. I will keep my score.

Review 3

Summary and Contributions: The paper considers the stochastic combinatorial multi-armed bandit problem (CMAB), where the learner can play a subset of arms and observes feedback on the arms they played. Motivated by the empirical efficacy of thompson sampling approaches in practice, the paper focuses on developing and analyzing a thompson sampling based approach for CMAB. 1. Assuming the reward distributions of individual arms are independent, the paper improves the regret bound for an existing TS based approach with Beta priors. 2. For the case when the reward distributions of different arms are correlated, the paper proposes a TS based approach with Gaussian priors and proves theoretical bounds.

Strengths: 1. The paper presents a Thompson Sampling (TS) algorithm for the CMAB problem when the rewards from different arms are correlated. Given that the correlated arms is realistic in many CMAB applications and TS is known for its empirical performance, this algorithm would be of larger interest. 2. The paper also improves bounds for existing algorithms in the case of arms having independent rewards.

Weaknesses: The technical contributions in this paper are marginal for a theoretical work. 1. The improvement in bounds for the TS-Beta algorithm are marginal and the techniques used doesn't apply to other problems. 2. Similarly, the algorithm presented for the case of correlated rewards is a natural generalization of well known algorithm for the stochastic MAB problem. I believe the second algorithm could be improved if we can sample from a multi-variate gaussian prior instead of the current choice of independent gaussian priors. 3. I think the paper can also improve the writing, particularly articulating the implications of correlated priors vs independent priors.

Correctness: Yes

Clarity: There's room for improvement in the writing. 1. Clearly articulating challenges involving in generalizing TS based approaches to correlated rewards setting and how the paper resolved them. 2. Discussing the implication of different priors (correlated vs independent) on performance (even if it's empirical).

Relation to Prior Work: Largely, the paper does a good job, but some of the issues with writing also apply to discussing prior work.

Reproducibility: Yes

Additional Feedback: Missing reference: Thompson Sampling for the MNL Bandit. This paper discusses some interesting algorithmic choices for a very stylized CMAB problem. In particular, this paper shows that there's value in sampling parameters using a common posterior distribution instead of the regular independent sampling. Post rebuttal: I've read author's response, other reviewers comments and have taken these into account to maintain my original score. It would be good if the paper can adapt some of these ideas and compare it with the performance of Algorithm-2.

Review 4

Summary and Contributions: This paper studies stochastic combinatorial multi-armed bandit with semi-bandit feedback (CMAB). This paper lists three kinds of settings and mainly focuses on setting (i) and (iii). This paper first improves the result of Wang and Chen [2018] by providing the regret upper bound for CTS-BETA in the setting (i) with bounded outcomes and it also holds for nonlinear reward functions. Moreover, the authors propose a new policy called CTS-GAUSSIAN and give a regret bound reducing to O(log^2(m) n log(T)/\Delta) for a diagonal sub-Gaussian matrix. When the reward is linear, this paper proposes CLIP CTS-GAUSSIAN and give an improved regret bound. Experiments on applications show the superiority of the policies.

Strengths: - The problem is well defined and has some prior results in the literature. This paper not only improves the prior results for setting (i) but also fill the blank for setting (iii). - The new assumptions introduced in this paper is natural and acceptable. - Besides the rigorously proved better theoretical guarantee, this paper also conducts experiment to compare with the baselines in two applications, which shows that the proposed policy has the better performance and less computational time.

Weaknesses: - Since there still exist some gaps between the upper bound and lower bound, do the proposed upper bounds tight?

Correctness: I didn't go through the proofs in appendix.

Clarity: Yes.

Relation to Prior Work: Yes. This paper clearly compares with the prior work, ie. Wang and Chen [2018].

Reproducibility: Yes

Additional Feedback: