NeurIPS 2020

An Asymptotically Optimal Primal-Dual Incremental Algorithm for Contextual Linear Bandits

Review 1

Summary and Contributions: The paper studies algorithms for the finite-armed linear bandit problem under which the leading order term of the asymptotic growth rate of regret is optimal. The asymptotic limits of these problems seems to be well understood by older work of Lai and Graves but has attracted renewed interest. The paper aims to improve previous asymptotically algorithms and analysis in settings with smaller time horizon. To do this, they reformulate the lower bound to better account for contexts with small probabilities, they use a primal dual algorithm based on UCBs and Lagrangian relaxation of the asymptotic lower bound, and they show the constant terms in regret scale at most linearly with problem parameters like the number of contexts.

Strengths: While the problem's large-deviation-style asymptotics are well understood, its difficult to translate those into practical algorithms and performance. This paper takes a step in that direction. Writing this paper took real technical mastery. It is quite complex and creatively combines many different ideas from the bandit and online optimization literatures.

Weaknesses: From a purely theoretical perspective, the paper provides a bound on the lower order asymptotic terms in an asymptotically optimal regret upper bound. But, it's not clear if these bounds are practically meaningful. First, the asymptotic bounds are pretty misleading for most linear bandit examples. The leading order term depends inversely on the smallest possible gaps between arms' mean rewards, which is often extremely small. Second, the bounds on the "lower order" terms are only smaller than the leading order term when log(n) exceeds the |X|d, meaning when the horizon is exponentially large in the number of contexts times the model dimension. The algorithm seems more practical than some previous ones that were asymptotically optimal in the same sense. But the experiments don't make a serious attempt to judge whether they could supplant e.g. Thompson sampling or a tuned UCB algorithm. The algorithm is inherently based around identifying an exactly optimal policy, even if other policies are extremely similar. I worry it could perform very poorly when several policies are statistically indistinguishable (which is very common).

Correctness: I don't see an clear issues with correctness, but the full paper consists of 49 extremely technical pages. Realistically, if accepted, the paper is being accepted without having being 'verified' in peer review.

Clarity: The body of the paper is relatively well written, though there is a way more material crammed in than fits in an eight page paper.

Relation to Prior Work: The paper should credit, from the very beginning, works like: Asymptotically Efficient Adaptive Choice of Control Laws in Controlled Markov Chains by Graves and Lai or Asymptotically efficient adaptive allocation schemes for controlled iid processes: Finite parameter space by A Rajeev, D Teneketzis, V Anantharam. Or even earlier work in experimental design that gives similar lower bounds. It should be clear, from the beginning, that the main ideas underlying asymptotic optimality are quite old are not a recent development in ML.

Reproducibility: Yes

Additional Feedback: Please clarify what parameters you used for LinUCB and LinTS? Are these the extremely conservative versions for which regret bounds exist, or the versions with more realistic parameters that are implemented in practice?

Review 2

Summary and Contributions: The paper proposes a novel algorithm for linear contextual bandits (with contexts in a finite support and finite number of actions). The paper is both asymptotically optimal and computationally efficient. The authors present as well the finite-time guarantee of the algorithm. Experiments, as well as theoretical statements, support this finding.

Strengths: The authors make a significant improvement on previous works (on constructing bandit algorithms with asymptotically optimal regret guarantee) by combining various techniques: - introducing a constraint in the optimization problem to guarantee bounded number of pulls - reformulating the optimization problem to better adapt to the context distribution - reformulating the Lagrangian dual formulation to an optimistic version, which removes requirement of wasteful forced exploration - introducing a new incremental learning method for scalability Each technique is well motivated, and described in a clear manner.

Weaknesses: -The paper seems to be much inspired by the work of Degenne et al. (2020). The authors do not explicitly state how their contributions differ from this work in the main body of the paper.

Correctness: The claims are clear. The empirical methodology is correct.

Clarity: The paper is well written and the exposition is clear. The contributions as well as how these were made are described in a clear manner.

Relation to Prior Work: The authors extensively review the prior works that address the same problem. The proposed method overcomes weakness of prior works in various aspects (computational efficiency, empirical performance, dependency on forced exploration and number of arms, dependency on the distribution of contexts)

Reproducibility: Yes

Additional Feedback: -In Preliminaries, the authors state that the Gaussian assumption on the errors can be relaxed to the sub-Gaussian assumption. It would be helpful know how the theoretical statements would change under sub-Gaussian assumption. (For example, can we still use the KL divergence formula as it is?) ==== Comments after rebuttal ==== Thank you for a detailed answer to my question. I also agree that an algorithm with low, asymptotic problem-dependent regret bound can be more practical than an algorithm with low, finite-time worst case regret bound.

Review 3

Summary and Contributions: Provided an asymptotically optimal algorithm for linear contextual bandits

Strengths: primal-dual algorithm applied to linear contextual bandits seems novel to me

Weaknesses: finite-time regret bound--the most important metric--is not good enough

Correctness: yes

Clarity: reasonable

Relation to Prior Work: yes Post rebuttal: I had read the authors' response and had decided to keep the current score.

Reproducibility: Yes

Additional Feedback: I read the paper with interest, but got a bit disappointed in the end. Asymptotic optimality seems to be the focus of the paper, and this is the point I disagree with. Certainly, having asymptotic optimality is good, but only performing well on that--rather than finite-time optimality--is not enough given that linear contextual bandits have been studied extensively. In particular, a simple epsilon-greedy algorithm with epsilon decreasing to 0 at an appropriate rate is already asymptotically optimal. So in my view, finite-time regret must be the clear performance metric for evaluating an algorithm. Additionally, the finite-time regret bound given in Theorem 2 is not very good. In particular, it has a dependence on the number of contexts: in the normal linear contextual bandits, there are infinite (and even uncountably infinite) contexts and hence the bound is vacuous except in the simplest setting. The authors need to take a look at "Contextual Bandits with Linear Payoff Functions/Chu, Li, Reyzin and Schapire, 2011", one of the pioneering papers (missing in the references too) that gave finite-time regret bound and established a \sqrt{dT} regret without any dependence on the number of contexts.

Review 4

Summary and Contributions: This papers presents an asymptotically optimal algorithm for Linear Contextual Bandits which improves performances compared to recent algorithms. The algorithms is based on a primal dual incremental implementation of a solution of the relaxed lower bound optimization problem. A finite time regret guarantee as well as empirical evaluations are provided. The authors use a new concentration inequality on the performance of the estimated regression parameter that can be of independent interest for the community.

Strengths: This paper improves recent algorithms on several aspects: - the context distribution is decoupled from the exploration policy, the regret bound is thus independent of the context distribution and in particular its minimum probability - allows a direct sampling from the exploration strategy A finite time regret bound is provided showing that the algorithm is asymptotically optimal and empirical evaluations show the improvement in performance compared to other recent algorithms. Finally the authors derive a new concentration inequality for the estimated regression parameters that can be of independent interest for the community.

Weaknesses: None as far I understand the main tools of the algorithm and the main results.

Correctness: I did not check the proofs of the main results

Clarity: I ernjoy reading the paper. It is well written and it is easy to understand the main tools and the main arguments for the suggested algorithm.

Relation to Prior Work: The authors clearly discussed the differences with previous contributions.

Reproducibility: Yes

Additional Feedback: l 15 : please give the chapter/section number when citing [1] l 120 : by -> be