Summary and Contributions: The paper deals with a variant of the dueling or battling bandit problem, where it is allowed to pull up to k many arms in each time step. In addition, a more general class of multiwise comparison models is considered as the ones investigated in the battling bandit problem or the previously used Multinomial Logit model of related settings. A lower bound on the expected regret is shown for the general class of multiwise comparison models, which in particular gives a more refined bound as the previously shown for the case of MNL models. Moreover, one learning algorithm is suggested for the considered problem and analyzed theoretically with respect to its expected regret bound. Finally, the proposed algorithm is investigated in numerous experiments on synthetic as well as real-world datasets and compared to related algorithms. Post Rebuttal After reading the other reviews as well as the author's response, I think that the authors have not addressed two of the concerns mentioned in the reviews thoroughly enough, namely the independence of k for the complexity terms as well as the suggestion to compare with the lower bounds coming from the structured bandits.

Operator learning has emerged as a new paradigm for the data-driven approximation of nonlinear operators. Despite its empirical success, the theoretical underpinnings governing the conditions for efficient operator learning remain incomplete. The present work develops theory to study the data complexity of operator learning, complementing existing research on the parametric complexity. We investigate the fundamental question: How many input/output samples are needed in operator learning to achieve a desired accuracy $\epsilon$? This question is addressed from the point of view of $n$-widths, and this work makes two key contributions. The first contribution is to derive lower bounds on $n$-widths for general classes of Lipschitz and Fr\'echet differentiable operators. These bounds rigorously demonstrate a ``curse of data-complexity'', revealing that learning on such general classes requires a sample size exponential in the inverse of the desired accuracy $\epsilon$. The second contribution of this work is to show that ``parametric efficiency'' implies ``data efficiency''; using the Fourier neural operator (FNO) as a case study, we show rigorously that on a narrower class of operators, efficiently approximated by FNO in terms of the number of tunable parameters, efficient operator learning is attainable in data complexity as well. Specifically, we show that if only an algebraically increasing number of tunable parameters is needed to reach a desired approximation accuracy, then an algebraically bounded number of data samples is also sufficient to achieve the same accuracy.

We consider the learning problem of finding a dependency between a general class of objects and another, possibly different, general class of objects. The objects can be for example: vectors, images, strings, trees or graphs. Such a task is made possible by employing similarity measures in both input and output spaces using ker(cid:173) nel functions, thus embedding the objects into vector spaces.