I'm trying to understand an Online Portfolio Selection (OPS) strategy proposed by Zhang et al. (2021), named combination weights based on online gradient descent (CW-OGD). The algorithm is as follows:
My question is about the 7th line of the algorithm. I do not know how to calculate the ${\nabla _t}$. Let's recap the essence of related variables:
$\begin{array}{l}j = \text{Index of Expert}\\k = \text{Number of Experts}\\t = \text{Index of Time Period}\\n=\text{Number of trading days}\\w_t^j = {\text{weight factor assigned to expert }}j{\text{ at time }}t\\K = \text{Weight Set} = \left\{ {\mathbf{w}:\sum\nolimits_{j = 1}^k {{w^j} = 1,\;{w^j} \in \left[ {0,1} \right]} } \right\}\\\gamma = \text{Scalar}\\\mathbf{w}=\left( {w_t^1,w_t^2, \ldots ,w_t^k} \right) \in {\mathbb{R}^k}\\\eta_t=\text{Scalar}\end{array}$
About the 7th line of the algorithm, the following explanation is provided in the paper:
The gradient upper bound of investor’s average loss function is $G$ for all $t$ and $\mathbf{w} \in K$, i.e., $G = {\sup _{t \in n,\mathbf{w} \in K}}\left\| {\nabla A{L_t}\left( \mathbf{w} \right)} \right\|$. Using the OGD algorithm to update the combination weights, the weight vector for the next period wt+1 can be calculated as
${\mathbf{w}_{t + 1}} = \prod\limits_K {\left( {{\mathbf{w}_t} - {\eta _t}{\nabla _t}} \right)}$
where ${{\nabla _t}}$ is the abbreviation of the gradient vector for average loss function $\nabla A{L_t}\left( {{\mathbf{w}_t}} \right)$. $\prod\limits_K {}$ is the projection under Euclidean norm, which pushes the update model to lie inside the feasible domain, i.e., $\prod\limits_K {\left( \mathbf{y} \right)} = \arg {\min _{\mathbf{x} \in K}}\left\| {\mathbf{x} - \mathbf{y}} \right\|$.
P.S.:
They did not mention the shape of $\mathbf{b}^j$, but I guess it's a vector of size $m$ (${{\bf{b}}^j} \in {\mathbb{R}^m}$ where $m$ is the number of stocks.)