Data-Dependent Bounds for Online Portfolio Selection Without Lipschitzness and Smoothness
This addresses the challenge of optimizing investment strategies in online convex optimization without smoothness assumptions, representing a foundational advance in the field.
The paper tackled the problem of online portfolio selection with non-Lipschitz, non-smooth losses by introducing the first data-dependent regret bounds, achieving sublinear worst-case rates and logarithmic regrets for easy data.
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.