Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond
This work addresses an open problem in online learning theory, providing improved algorithms for dynamic regret minimization with applications in sequential decision-making, though it is incremental relative to existing frameworks.
The paper tackles the problem of achieving optimal dynamic regret in proper online learning with strongly convex losses, showing that Strongly Adaptive algorithms can achieve near-optimal dynamic regret of $ ilde O(d^{1/3} n^{1/3} ext{TV}[u_{1:n}]^{2/3} \vee d)$, improving upon prior work by handling non-smooth losses and enhancing dimension dependence.
We study the framework of universal dynamic regret minimization with strongly convex losses. We answer an open problem in Baby and Wang 2021 by showing that in a proper learning setup, Strongly Adaptive algorithms can achieve the near optimal dynamic regret of $\tilde O(d^{1/3} n^{1/3}\text{TV}[u_{1:n}]^{2/3} \vee d)$ against any comparator sequence $u_1,\ldots,u_n$ simultaneously, where $n$ is the time horizon and $\text{TV}[u_{1:n}]$ is the Total Variation of comparator. These results are facilitated by exploiting a number of new structures imposed by the KKT conditions that were not considered in Baby and Wang 2021 which also lead to other improvements over their results such as: (a) handling non-smooth losses and (b) improving the dimension dependence on regret. Further, we also derive near optimal dynamic regret rates for the special case of proper online learning with exp-concave losses and an $L_\infty$ constrained decision set.