__ Summary and Contributions__: This paper considers the implicit update algorithm for online learning. It is well known that this algorithm works very well in practice, although theoretical understanding of its benefits over online mirror descent are somewhat lacking. This paper shows that the implicit update algorithm enjoys a regret bound that is adapted to the variability of the sequence of loss functions. This regret bound holds even if the learning rate used is a constant. At the same time, the method has a worst case regret bound of O(sqrt{T}) by choosing a learning rate of O(1/sqrt{T}). To overcome this tuning of the learning rate, the authors also give an adaptive version of the implicit update method, called AdaImplicit, which uses techniques from the work of Orabona and Pal (2015) to careful change learning rates and automatically achieve the required regret bounds without knowledge of the variability of the loss functions. Crucially, this algorithm doesn't make use of the doubling trick, which also gives the same theoretical bound, but is worse practically. Finally, the authors conduct experiments showing the superior performance of the method over a wide range of hyperparameters.

__ Strengths__: + The paper is well-written, with a careful discussion of the method, the analysis, and prior work.
+ The result itself is quite nice, and sheds some light on why implicit methods can be expected to outperform standard online mirror descent methods.
+ The AdaImplicit algorithm updates the learning rates in an adaptive fashion leading to the right regret bounds without any knowledge of the variability, and does so in a practical manner (i.e. without using the doubling trick).
+ Experiments show that AdaImplicit has good performance and is not sensitive to the tuning of its one parameter, beta.

__ Weaknesses__: - The main weakness as far as I can see is that the same sort of regret bound in terms of the variability of the sequence can be obtained much more easily for the "lazy" version of implicit updates, i.e. a follow-the-regularized-leader style algorithm which computes x_t by minimizing \sum_{s < t} \ell_s(x) + \lambda \Psi(x). This is simply because of the follow-the-leader/be-the-leader inequality. Could the authors comment on this aspect?
- One major issue with implicit update methods is that unless there is some special structure, the update requires an "inner loop" of optimization to compute. The authors do mention this point briefly, but a more detailed discussion of this, especially in the context of practical implementations, would be useful.
Post rebuttal comments: I agree with the authors that the lazy version of FTRL that I described would be harder to implement in practice. This makes their result more compelling.

__ Correctness__: The claims seem to be correct to the extent I checked the details. I did not verify all the proofs in the appendix though. The experimental methodology is quite sound.

__ Clarity__: The paper is very well-written.

__ Relation to Prior Work__: Comparison with prior work is done well in this paper.

__ Reproducibility__: Yes

__ Additional Feedback__: Post rebuttal:
Thank you, I agree that lazy FTRL would be harder to implement in a practical setting, even though it would give the same temporal variability bound. If accepted, please add this point to the paper. Also, some more discussion on implementing the implicit update (as mentioned in the rebuttal) would be useful to have in the next version of the paper.

__ Summary and Contributions__: The paper presents algorithms for the online convex optimisation framework with the upper bound formulated in terms of temporal variability, i.e., the sum of maximum differences of \ell_t(x) - \ell_{t-1}(x) over time (I am actually surprised there is no absolute value around the difference in (1)). If temporal variability vanishes, the regret defaults to O(\sqrt{T}). Theorem 6.3 shows optimality of the bound, in the sense that the algorithm of the paper can be made to suffer particular regret.

__ Strengths__: The paper provides an interesting answer to a very natural question.

__ Weaknesses__: Of course, one cannot help asking if the lower bound extends to an arbitrary algorithm. One expects it should...

__ Correctness__: Please confirm there is no absolute value in (1). Is this correct?

__ Clarity__: Yes

__ Relation to Prior Work__: Yes.
It would be interesting to know what the result of the paper implies for the changing dependency framework in prediction with expert advice (E. Moroshko, N. Vaits, and K. Crammer. Second-order non-stationary online learning for regression. Journal of Machine Learning Research, 2015; Y. Kalnishkan, An Upper Bound for Aggregating Algorithm for Regression with Changing Dependencies, 2016).

__ Reproducibility__: Yes

__ Additional Feedback__: Thank you for the answer to my comments. I am keeping my score.

__ Summary and Contributions__: This paper develops a new analysis framework of online mirror descent using implicit updates, and derives a static regret bound in terms of the variability of loss functions $V_T$. Then a refined version of the implicit algorithm is proposed using adaptive learning rates, which is then proved able to meet the lower bound in settings where “loss functions vary slowly”.

__ Strengths__: This paper considers a new setting that lies between the static setting and the dynamic setting (i.e., the loss functions only vary slowly). Although it is not presented clear enough in this paper, such setting will possibly provide a new vision in this area and may be useful in some specific scenarios.

__ Weaknesses__: This paper does not present its learning setting clear enough. In general, this work seems to be conducted in static setting, as the learning objective is the static regret. However, the bound is derived in terms of the variability of loss functions $V_T$, which is a characteristic in dynamic setting. Throughout the paper the authors claim that this work is more suitable to “small” V_T, but without further explanation or justification. I then wonder what is “small” V_T (this even contradict the experiment setup where “there is no reason to believe the temporal variability is small”).
This work has a strong overlap with previous work (FIOL [1]), thus the contribution and novelty is very limited. More specifically, the algorithm seems to be a special case of FIOL which omits the regularization term, and the theoretical analysis is also very similar (see my detailed remarks below). The main novelty in theoretical analysis is to use $V_T$ to bound the gain, which is quite straightforward to deduce. Using adaptive learning rate to improve the algorithm also seems to be normal, as it is directly borrowed from AdaFTRL.
[1] C. Song, J. Liu, H. Liu, Y. Jiang, and T. Zhang. Fully implicit online learning.

__ Correctness__: Yes. They are correct.

__ Clarity__: Yes.

__ Relation to Prior Work__: This paper does not discuss its relation with prior works in the dynamic online setting where the variability $V_T$ is also a crucial characteristic to measure how drastic the environment is varying.

__ Reproducibility__: Yes

__ Additional Feedback__: Typo: Line 32, choose to not use -> choose not to use.
（update after rebuttal）
After reading the response, I understand the setup, and acknowledge the contribution as the first work to link the regret of implicit updates to the variability of losses and further exploit this quantity to attain a smaller regret. However, I still think the overlaps with previous works are somehow strong, which affects the novelty of this paper. Specifically, the designed algorithm is actually a special case of FIOL, and theoretical derivations for the step-wise lemma (Lemma 1) and the regret bound are essentially the same as those in [18].
Given these points, I decide to raise my score to borderline reject.