Sparsity-Based Interpolation of External, Internal and Swap Regret

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 Tgg d rounds in total, we present a single algorithm achieving the instance-adaptive φ-regret bound beginequation* tilde Oleft(minleft√d-dunifφ+1,√d-dselfφright·√Tright), endequation* where dunifφ is the maximum amount of experts modified identically by φ, and dselfφ is the amount of experts that φ trivially modifies to themselves. By recovering the optimal O(√Tlog d) external regret bound when dunifφ=d, the standard tilde O(√T) internal regret bound when dselfφ=d-1 and the optimal tilde O(√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.