  This paper introduces a novel family of generalized exponentiated gradient
(EG) updates derived from an Alpha-Beta divergence regularization function.
Collectively referred to as EGAB, the proposed updates belong to the category
of multiplicative gradient algorithms for positive data and demonstrate
considerable flexibility by controlling iteration behavior and performance
through three hyperparameters: $\alpha$, $\beta$, and the learning rate $\eta$.
To enforce a unit $l_1$ norm constraint for nonnegative weight vectors within
generalized EGAB algorithms, we develop two slightly distinct approaches. One
method exploits scale-invariant loss functions, while the other relies on
gradient projections onto the feasible domain. As an illustration of their
applicability, we evaluate the proposed updates in addressing the online
portfolio selection problem (OLPS) using gradient-based methods. Here, they not
only offer a unified perspective on the search directions of various OLPS
algorithms (including the standard exponentiated gradient and diverse
mean-reversion strategies), but also facilitate smooth interpolation and
extension of these updates due to the flexibility in hyperparameter selection.
Simulation results confirm that the adaptability of these generalized gradient
updates can effectively enhance the performance for some portfolios,
particularly in scenarios involving transaction costs.
