Unconstrained Online Optimization: Dynamic Regret Analysis of Strongly Convex and Smooth Problems
This work addresses dynamic regret bounds in online learning, offering incremental improvements in algorithm efficiency and regret guarantees for optimization problems.
The paper tackles unconstrained online optimization for strongly convex and smooth functions, showing that a preconditioned OGD variant achieves O(C*_{2,T}) regret with one gradient query per round, and proposing an OON method that achieves O(C*_{4,T}) regret when first and second order information is predictable, which can be much smaller.
The regret bound of dynamic online learning algorithms is often expressed in terms of the variation in the function sequence ($V_T$) and/or the path-length of the minimizer sequence after $T$ rounds. For strongly convex and smooth functions, , Zhang et al. establish the squared path-length of the minimizer sequence ($C^*_{2,T}$) as a lower bound on regret. They also show that online gradient descent (OGD) achieves this lower bound using multiple gradient queries per round. In this paper, we focus on unconstrained online optimization. We first show that a preconditioned variant of OGD achieves $O(C^*_{2,T})$ with one gradient query per round. We then propose online optimistic Newton (OON) method for the case when the first and second order information of the function sequence is predictable. The regret bound of OON is captured via the quartic path-length of the minimizer sequence ($C^*_{4,T}$), which can be much smaller than $C^*_{2,T}$. We finally show that by using multiple gradients for OGD, we can achieve an upper bound of $O(\min\{C^*_{2,T},V_T\})$ on regret.