This work introduces the first small-loss and gradual-variation regret bounds for online portfolio selection, marking the first instances of data-dependent bounds for online convex optimization with non-Lipschitz, non-smooth losses. The algorithms we propose exhibit sublinear regret rates in the worst cases and achieve logarithmic regrets when the data is "easy," with per-iteration time almost linear in the number of investment alternatives. The regret bounds are derived using novel smoothness characterizations of the logarithmic loss, a local norm-based analysis of following the regularized leader (FTRL) with self-concordant regularizers, which are not necessarily barriers, and an implicit variant of optimistic FTRL with the log-barrier.

翻译：本文首次提出在线投资组合选择的小损失和渐进变分遗憾界，标志着非Lipschitz、非光滑损失函数下的在线凸优化首次实现数据依赖界。我们提出的算法在最坏情况下具有次线性遗憾率，在数据较“简单”时达到对数遗憾，且每次迭代的时间复杂度几乎与投资选项数量成线性关系。这些遗憾界通过以下方法推导得出：对数损失函数的新颖光滑性刻画、基于局部范数分析的自正则跟随正则化领导者（FTRL）方法（采用不必为障碍函数的自和谐正则化器），以及基于对数障碍函数的隐式乐观FTRL变体。