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.

翻译：本文提出了一族基于Alpha-Beta散度正则化函数推导的新型广义指数梯度（EG）更新方法。该系列方法统称为EGAB，属于针对正数据的乘法梯度算法类别，通过三个超参数（α、β和学习率η）控制迭代行为与性能，展现出显著的灵活性。为在广义EGAB算法中对非负权重向量实施单位l₁范数约束，我们开发了两种略有差异的实现方案：一种利用尺度不变损失函数的特性，另一种则依赖于可行域上的梯度投影。为展示其适用性，我们通过基于梯度的方法评估了所提更新规则在在线投资组合选择问题中的表现。这些方法不仅为各类OLPS算法（包括标准指数梯度与多种均值回归策略）的搜索方向提供了统一的理论视角，其超参数选择的灵活性还支持对这些更新规则进行平滑插值与扩展。仿真实验证实，这些广义梯度更新的自适应特性能够有效提升部分投资组合的性能，尤其在涉及交易成本的场景中表现突出。