Update equation for the hidden-to-output weights

The input of $j$-th neuron on $m$-th panel in the output layer is obtained as

$u_{j, m} = u_j = \bar{v}_{w_j}' \cdot \bar{h} = \sum_{i = 1}^N w_{ij}' h_i = \sum_{i = 1}^N w_{ij}' w_{ki} \pod{\text{1}}$

and the derivative of $E$ to $u_{j,m}$ is

$\frac{\partial E}{\partial u_{j,m}} = y_{j,m} - \delta_{j j_{o,m}} \doteq e_{j,m} \pod{\text{2}}$

By using $(1)$ and $(2)$, the derivative of $E$ to $w_{ij}'$ is obtained as

$\frac{\partial E}{\partial w_{ij}'} = \sum_{m = 1}^C \frac{\partial E}{\partial u_{j,m}} \frac{\partial u_{j,m}}{\partial w_{ij}'} = \sum_{m = 1}^C e_{j,m} h_i \pod{\text{3}}$

by which, the update equation for the hidden-to-output weights is

${w_{ij}'}^{(new)} = {w_{ij}'}^{(old)} - \eta \frac{\partial E}{\partial w_{ij}'} = {w_{ij}'}^{(old)} - \eta \sum_{m = 1}^C e_{j,m} h_i \pod{\text{4}}$

which can also be represented as

${\bar{v}_{w_j}'}^{(new)} = {\bar{v}_{w_j}'}^{(old)} - \eta \sum_{m = 1}^C e_{j,m} \bar{h} = {\bar{v}_{w_j}'}^{(old)} - \eta \sum_{m = 1}^C e_{j,m} \bar{v}_{w_k} , \ j = 1, 2, \cdots, V \pod{\text{5}}$

where the prediction error $e_{j,m}$ is summed across all context words in the output layer.
Note that it is needed to apply the update equation for every hidden-to-output weight for each training instance.

Update equation for input-to-hidden weights

The output of the hidden layer is

$h_i = v_{w_k, i} = w_{ki} \pod{\text{6}}$

By using $(1)$ and $(2)$, the derivative of $E$ to $h_i$ is obtained as

$\frac{\partial E}{\partial h_i} = \sum_{j = 1}^V \sum_{m = 1}^C \frac{\partial E}{\partial u_{j,m}} \frac{\partial u_{j,m}}{\partial h_i} = \sum_{j = 1}^V \sum_{m = 1}^C e_{j,m} w_{ij}' \pod{\text{7}}$

By using $(6)$ and $(7)$, we can obtain the derivative of $E$ to $w_{ki}$ as

$\frac{\partial E}{\partial w_{ki}} = \frac{\partial E}{\partial h_i} \frac{\partial h_i}{\partial w_{ki}} = \sum_{j = 1}^V \sum_{m = 1}^C e_{j,m} w_{ij}' \pod{\text{8}}$

Thus, the update equation for the input-to-hidden weights is

$w_{ki}^{(new)} = w_{ki}^{(old)} - \eta \frac{\partial E}{\partial w_{ki}} = w_{ki}^{(old)} - \eta \sum_{j = 1}^V \sum_{m = 1}^C e_{j,m} w_{ij}' \pod{\text{9}}$

or equivalently

$\bar{v}_{w_k}^{(new)} = \bar{v}_{w_k}^{(old)} - \eta \sum_{j = 1}^V \sum_{m = 1}^C e_{j,m} \bar{v}_{w_j}' \pod{\text{10}}$