__ Summary and Contributions__: This paper deals with the problem of online convex optimization with hard switching costs: the online learning algorithm must not change the action more than K times. The authors focus on the case where the adversary selects only linear functions.
The contribution of the paper is threefold:
- A lower bound on the regret based on applying Abernethy's adversary (for the lower bound in regret in OCO) when the agent changes her action.
- An upper bound on the regret based on taking batches of T/K consecutive epochs and applying stochastic gradient in the now K-horizon problem. This gives a matching bound on the order of K and T and exact for dimension n> 1
- An exact lower bound for n=1: this is obtained via a minmax analysis of a novel type of game ("fugal game").

__ Strengths__: 1. This is the first paper to provide regret bounds (which also happen to be tight) with a constrained number of switches rather than just switching costs.
2. The fugal game (to obtain bounds for n =1 ) is a quite neat idea.

__ Weaknesses__: 1. The techniques used to obtain the bounds for n > 1 are rather standard. It is also not clear if the fugal game can be extended/used for other cases.

__ Correctness__: The paper is a theoretical one, and the proofs and claims seem correct.

__ Clarity__: The paper is generally well written, however the notions of adaptive and oblivious adversary are not well explained before (it is unclear if the adaptive adversary chooses the function only or her action is dynamic as well) they are used in the introduction.

__ Relation to Prior Work__: Work related to learning with switching costs is adequately discussed and compared with.
However, there is a small issue related Cover's counterexample and the notion of regret. Cover's paper says that an adversary can force O(T) regret if she can change her action at each decision epoch; regret as defined before forces the adversary to pick a static action, so the above result does not apply.

__ Reproducibility__: Yes

__ Additional Feedback__: The authors mention that, in the studied case with continuous action spaces, no phase transition is observed; this is an interesting observation, however it would be nice to have some intuitive explanation why this is the case and what are the properties of this setting that allow this "no phase transition".
------- Update after the authors' reply ---------
I would like to thank the authors for their clarifications, though it is still not clear if the fugal game can be used outside this specific case. Overall it is a solid paper with a neat result, my score remains the same.

__ Summary and Contributions__: This paper considers the problem of online convex optimization, in a scenario where the learning algorithm can only change its action K < T times. This is a natural scenario for many online decision-making problems. For instance if the current state x_i corresponds to the parameter settings of some software system and changing the action corresponds to pushing out a new update of the system, or if x_i corresponds to an allocation of an R&D budget within a company and changing the action means modifying how the R&D budget is allocated, then you would want to limit the number of times you need to do that. This type of problem has been considered extensively in the classic discrete-action combining-expert-advice scenario, and there is some past work in the OCO setting, but this paper gets tight minimax bounds. Interestingly, it shows that unlike the discrete case, in the OCO case there is no phase transition. Also it defines an interesting “fugue” game to tightly analyze the 1-dimensional case even up to constants. Overall, a good and substantial paper.

__ Strengths__: The paper achieves tight minimax bounds for a natural online problem. It also shows that unlike in the discrete “combining expert advice” problem, there is no phase transition. Analysis is interesting and substantial.

__ Weaknesses__: The space this is operating in is a little crowded - there are a variety of results over the years and they depend on specific assumptions of the problem setup.

__ Correctness__: Yes, as far as I can tell

__ Clarity__: Yes, the paper is well written.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: Can you explain more what is the reason for the disappearance of the phase transition compared to the discrete game? In the discrete game without a switching bound, you would typically get a total cost of (1+epsilon)*OPT + (log n)/epsilon, and then set epsilon = sqrt((log n)/T) to balance the losses. This means that even without a switching bound, against an oblivious adversary, you’re switching between actions only about 1/sqrt(T) of the time anyway. On the other hand, you are constantly modifying your probability distribution every round (just that this change is only occasionally reflected in a switch of actions). In your game, there isn’t this distinction between the hidden probability distribution and the action played. Is that partly the difference? It would be great to have more discussion of this.
Also, is there a fundamental difference between limiting the number of switches and having a cost per switch? I couldn’t get a sense of that from the discussion in appendix A. In relation to the Metrical Task System problem, my intuition is the “who goes first” aspect is minor, since once you have costs to switch, it doesn’t really hurt the adversary much to reveal its loss function first, so long as losses per round are small – the big difference is the quantity being compared against. Another (old) paper perhaps worth relating to is: Baruch Awerbuch, Yossi Azar, Amos Fiat, Frank Thomson Leighton: Making Commitments in the Face of Uncertainty: How to Pick a Winner Almost Every Time, STOC 1996, which analyzes limited switching from a multiplicative ratio perspective.
Note added: Thanks to the authors for their informative response and clarifications.

__ Summary and Contributions__: This paper studies switching-constrained online convex optimization, where the learner can switch her action at most K times. The authors propose a mini-batch-paradigm-based online gradient descent algorithm for solving this problem, which is proved to enjoy an O(T/\sqrt{K}) regret bound. They also provide a matching lower bound for this problem.

__ Strengths__: 1. The problem is interesting and well-motivated. Optimization under Constrained switching cost widely exists in real-world applications. It has been extensively studied in both learning with experts and the bandits setting, but rarely considered in the online convex optimization paradigm.
2. This paper successfully established the minimax bound for this problem, and proposed a mini-batching algorithm to achieve a matching upper bound.

__ Weaknesses__: Novelty:
1. One of the main contribution of the paper is the algorithm proposed in Section 5 which achieves the minimax optimal regret bound. However, the main technique used here is the mini-batch paradigm which has been considered before, e.g., in [Dekel et al., 2011; Arora et al., 2012], and it seems that the proof technique is rather straightforward.
2. It seems that the adversary’s strategy and the minimax analysis in Section 4.1 are largely based on those of [Abernethy et al., 2008]. It would be better if the authors could add more discussions on what are the difficulties to adapt the minimax analysis for classic OCO into the switch-constrained setting.
Significance:
2. The fugal game proposed in this paper is novel and very interesting, but I am unsure about the significance of this contribution since it is considered in the 1-d situation and it only improves the minimax bound by constant factors.
--------------------------------------Post Rebuttal---------------------------
The authors cleared my concerns in the rebuttal. I am happy to raise my score.

__ Correctness__: I have read the main paper and made high level checks of the proofs, and I didn’t find any significant errors.

__ Clarity__: The paper is generally well-written and structured clearly.

__ Relation to Prior Work__: The relation to prior work is clearly discussed.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The authors discuss the problem of OCO with limited switching. They prove minimax lower bounds and upper bounds, with sharp constants in one dimension and higher dimensions. They introduce a new novel approach, fugal games, to prove sharp constants for minimax lower bound of one dimensional problem.

__ Strengths__: The theory is nicely presented, and the proofs are adequate and easy to follow (except the proofs for one-dimensional case, which I found very hard to follow and read; however, it is indeed the case when one tries to find a sharp constant for these types of problems). The contribution sits among other papers (such as "Consistent Online Learning: Convex and Submodular" and "Online learning over a finite action set with limited switching", and sheds light on the difference between convex set of actions and finite action set.

__ Weaknesses__: There are no experiments, but I do not think it is a problem, as it is a theory paper, and I liked it that the authors decided to put parts of the proof in the main paper.
The part of the paper discussing the relations with "Consistent Online Learning: Convex and Submodular" paper is not completely correct. What they prove in the paper is more general than that and the authors here decided to put a corrolary instead of the general result. However, this is understandable, as the main focus of this paper is minimax regret, not expected regret.

__ Correctness__: As far as I read, and skimmed through parts of the appendix, the arguments seems to be correct. However, I did not read the one-dimensional case completely, as it was very technical and hard to follow.

__ Clarity__: Yes, I enjoyed reading it.

__ Relation to Prior Work__: Yes, as far as I know, the results are completely compared.

__ Reproducibility__: Yes

__ Additional Feedback__: I have read the authors' feedback, as well as other reviewers' comments, and still think that the score given is well adjusted.