Power of Hints for Online Learning with Movement Costs

We consider the online linear optimization problem with movement costs, a variant of online learning in which the learner must not only respond to cost vectors $c_t$ with points $x_t$ in order to maintain low regret, but is also penalized for movement by an additional cost $\|x_t-x_{t+1}\|^{1+\epsilon}$ for some $\epsilon>0$. Classically, simple algorithms that obtain the optimal $\sqrt{T}$ regret already are very stable and do not incur a significant movement cost. However, recent work has shown that when the learning algorithm is provided with weak ``hint'' vectors that have a positive correlation with the costs, the regret can be significantly improved to $\log(T)$. In this work, we study the stability of such algorithms, and provide matching upper and lower bounds showing that incorporating movement costs results in intricate tradeoffs between $\log(T)$ when $\epsilon\ge 1$ and $\sqrt{T}$ regret when $\epsilon=0$.