Paper:
Adaptive Online Learning in Dynamic Environments
In this paper, we study online convex optimization in dynamic environments, and aim to bound the dynamic regret with respect to any sequence of comparators. Existing work have shown that online gradient descent enjoys an O(
where T is the number of iterations and P T is the path-length of the comparator sequence. However, this result is unsatisfactory, as there exists a large gap from the Ω( T (1 + P T )) lower bound established in our paper. To address this limitation, we develop a novel online method, namely adaptive learning for dynamic environment (Ader), which achieves an optimal O( T (1 + P T )) dynamic regret. The basic idea is to maintain a set of experts, each attaining an optimal dynamic regret for a specific path-length, and combines them with an expert-tracking algorithm. Furthermore, we propose an improved Ader based on the surrogate loss, and in this way the number of gradient evaluations per round is reduced from O(log T ) to 1. Finally, we extend Ader to the setting that a sequence of dynamical models is available to characterize the comparators.

Review:
This paper studies online convex optimization in dynamic environments, where one wishes to control the regret with respect to sequences of comparators, as opposed to a static comparator. The complexity of a comparator sequence is measured by its path length P_T, and the goal is to obtain an optimal regret bound simultaneously for all path lengths.  A first bound in this setting was established in the pioneering paper [1], which showed that online gradient descent (OGD, projected on the convex compact domain) with step-size 1/sqrt(T) achieves an at most T^{1/2} (1 + P_T) regret for all comparator sequences. However, there is a gap between this upper bound and the (T (1+P_T) )^{1/2} lower bound on worst-case regret established in Theorem 2 of the present paper, which is the first lower bound for this problem. On the other hand, if the path length P_T one wishes to compare against is known in advance, optimally tuning the step-size of OGD with respect to P_T in the OGD regret bound yields optimal (T (1+P_T) )^{1/2} regret for this path length.  Building on this observation, two algorithms are introduced which satisfy (T (1+P_T))^{1/2} regret simultaneously for all path lengths P_T, matching the above lower bound, up to a doubly logarithmic term in T. The first algorithm, called Ader, consists of a master which aggregates O(log T) instances of OGD with step-sizes in a geometric grid, using an expert algorithm (namely exponential weights). The standard expert regret bound implies a regret bound with respect to all step-sizes, and hence adaptivity to P_T (Theorem 3). As mentioned in the paper, the idea of aggregating different instances of OGD on a grid of step-sizes was considered in [2], to obtain different kinds of adaptivity guarantees. Note that Ader requires to query O(log T) values and gradients at each step, while OGD only needs 1 gradient. To address this, the authors introduce a variant of their algorithm, called "improved Ader", which consists of Ader run on linearized surrogate losses based on the gradient at the point chosen by the master algorithm (hence, only this gradient is used). Using the standard "gradient trick" (regret with respect to the original loss functions is bounded by regret with respect to the linearized losses) and the regret bound for Ader, a similar optimal (T (1+P_T))^{1/2} regret bound is obtained for improved Ader (Theorem 3). Finally, the algorithm and results are adapted to the setting when one measures the complexity of the sequence in terms of deviation with respect to a dynamical model rather than path length, introduced in [3].  This paper is superbly written and easy to follow; in particular, the "Related Work" section provides a clear and detailed account of existing work on the subject. The lower bound and adaptive upper bound for this problem are novel. The approach is natural and the techniques used somewhat straightforward, which I would argue is a plus rather than a minus.  Note: The upper and lower bounds are almost matching, up to a log log T factor. While this additional factor can be safely ignored as almost constant, it turns out that it can be avoided altogether, as follows. Instead of putting a uniform prior on the exponential grid of step-sizes, set a mass 1/(i(i+1)) to the i-th step-size (with i as in Eq. (9)), and set the expert learning rate to alpha = 1/T^{1/2}. The regret bound for the exponential weights wrt i is of order T^{1/2} log (i(i+1)) [more sophisticated experts algorithms would yield (T log (i(i+1)))^{1/2} regret wrt to i, but this does not bring any improvement here], which after substituting for the optimal i^* = O(log P_T) for a given value of P_T, gives a total O( (T (1+P_T))^{1/2} + T^{1/2} log log P_T ) = O ((T (1+P_T))^{1/2}) universal dynamic regret bound.  A few minor corrections: - Line 113: "an" --> "a"; - Line 142: "holds" --> "to hold"; - Line 171: "setting" --> "set"; - It may be worth adding, probably around or after Eq (10) and (11), that regret wrt true losses f_t is lower than regret wrt surrogate losses \ell_t, with a line that simply shows it (e.g., f_t (x_t) - f_t (u_t) \leq \ell_t (x_t) - \ell_t (u_t) from (10) and (11)). Although, the current formulation is already clear and the trick is simple and standard enough; - Line 202: a reference to the relevant section and theorem can be added.   [1] M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on Machine Learning (ICML), pages 928–936, 2003.  [2] T. van Erven and W. M. Koolen. Metagrad: Multiple learning rates in online learning. In Advances in Neural Information Processing Systems 29, pages 3666–3674, 2016.  [3] E. C. Hall and R. M. Willett. Dynamical models and tracking regret in online convex programming. In Proceedings of the 30th International Conference on Machine Learning, pages 579–587, 2013.