We propose a new class of online learning algorithms, generalized implicit Follow-The-Regularized-Leader (FTRL), that expands the scope of FTRL framework. Generalized implicit FTRL can recover known algorithms, as FTRL with linearized losses and implicit FTRL, and it allows the design of new update rules, as extensions of aProx and Mirror-Prox to FTRL. Our theory is constructive in the sense that it provides a simple unifying framework to design updates that directly improve the worst-case upper bound on the regret. The key idea is substituting the linearization of the losses with a Fenchel-Young inequality. We show the flexibility of the framework by proving that some known algorithms, like the Mirror-Prox updates, are instantiations of the generalized implicit FTRL. Finally, the new framework allows us to recover the temporal variation bound of implicit OMD, with the same computational complexity.

翻译：我们提出了一类新的在线学习算法——广义隐式跟随正则化领导算法（Generalized Implicit Follow-The-Regularized-Leader, FTRL），该算法扩展了FTRL框架的适用范围。广义隐式FTRL不仅能够恢复已知算法（如带线性化损失的FTRL和隐式FTRL），还可设计新更新规则（如aProx和Mirror-Prox在FTRL中的扩展）。本文理论具有建设性，提供了统一简洁的框架来设计能直接改进遗憾最坏情况上界的更新方法。核心思想在于用Fenchel-Young不等式替代损失函数的线性化过程。我们通过证明Mirror-Prox等已知算法是广义隐式FTRL的特例，展示了该框架的灵活性。最后，新框架在保持相同计算复杂度的前提下，实现了隐式OMD的时间变化界恢复。