Zhou Lu, Y. Jennifer Sun, Zhiyu Zhang
Focusing on the expert problem in online learning, this paper studies the interpolation of several performance metrics via $φ$-regret minimization, which measures the total loss of an algorithm by its regret with respect to an arbitrary action modification rule $φ$. With $d$ experts and $T\gg d$ rounds in total, we present a single algorithm achieving the instance-adaptive $φ$-regret bound \begin{equation*} \tilde O\left(\min\left\{\sqrt{d-d^{\mathrm{unif}}_φ+1},\sqrt{d-d^{\mathrm{self}}_φ}\right\}\cdot\sqrt{T}\right), \end{equation*} where $d^{\mathrm{unif}}_φ$ is the maximum amount of experts modified identically by $φ$, and $d^{\mathrm{self}}_φ$ is the amount of experts that $φ$ trivially modifies to themselves. By recovering the optimal $O(\sqrt{T\log d})$ external regret bound when $d^{\mathrm{unif}}_φ=d$, the standard $\tilde O(\sqrt{T})$ internal regret bound when $d^{\mathrm{self}}_φ=d-1$ and the optimal $\tilde O(\sqrt{dT})$ swap regret bound in the worst case, we improve upon existing algorithms in the intermediate regimes. In addition, the computational complexity of our algorithm matches that of the standard swap-regret minimization algorithm due to (Blum and Mansour, 2007). Technically, building on the well-known reduction from $φ$-regret minimization to external regret minimization on stochastic matrices, our main idea is to further convert the latter to online linear regression using Haar-wavelet-inspired matrix features. Then, by associating the complexity of each $φ$ instance with its sparsity under the feature representation, we apply techniques from comparator-adaptive online learning to exploit the sparsity in this regression subroutine.