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：本文首次提出了在线投资组合选择的小损失和渐进变化遗憾界，标志着在非Lipschitz、非光滑损失条件下在线凸优化的首个数据相关界限实例。我们提出的算法在最坏情况下具有次线性遗憾率，当数据“简单”时能实现对数遗憾，且每轮迭代时间几乎与投资选项数量成线性关系。这些遗憾界是通过以下创新方法推导得出的：对数损失的新型光滑性刻画、基于局部范数的带自和谐正则化（FTRL）分析（该正则化器不一定是障碍函数），以及使用对数障碍函数的隐式乐观FTRL变体。