Generalized Exponentiated Gradient Algorithms and Their Application to On-Line Portfolio Selection

TL;DR

This paper introduces EGAB, a generalized family of exponentiated gradient updates derived from Alpha-Beta divergences, enabling flexible interpolation between additive and multiplicative updates via hyperparameters $(\alpha,\beta,\eta)$. It develops unnormalized (EGAB-U) and normalized (EGAB-N, EGAB-P) variants to handle nonnegativity and unit $\ell_1$-norm constraints, with scale-invariant and projection-based normalizations. The AB-divergence framework unifies and extends known gradient methods, and the OLPS case shows how transaction costs can be incorporated into the training objective, yielding more robust performance. Extensive OLPS experiments across 16 datasets demonstrate that learned EGAB configurations, particularly EGAB-P, offer competitive or superior performance and better turnover properties as trading costs rise, highlighting practical impact for real-time portfolio management.

Abstract

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: $α$, $β$, and the learning rate $η$. 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.

Figures (4)

• Figure 1: Plots of the generalized exponential function $\exp_{1-\beta}(x)$ for different values of parameter $\beta$.
• Figure 2: Visualization, in three dimensions, of the $l_1$-norm constraint for nonnegative vectors. In blue color it is shown the orthogonal vector to the manifold which intersects with it at the uniform portfolio $\hbox{\boldmath $u$} \dot=\tfrac{1}{N}{\bf 1}$.
• Figure 3: Illustration of the decrease in geometric mean of cumulative wealth (over all the datasets) with the increase of the commission rate.
• Figure 4: Boxplots of the turnover of the algorithms, for the NYSE-N dataset, when varying the commission rate $c_r\in\{0\%,0.025\%,0.25\%\}$. The figure illustrates how the proposed algorithms (in blue) progressively lower their turnover when commissions rise.