Project: s41467-024-46413-6 | Function: update (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Implements the VREx (Variance Risk Extrapolation) domain generalization algorithm with risk extrapolation loss function for Cancer-Finder training

**KEY TERMS:** {
  "penalty_weight": "� hyperparameter in risk extrapolation",
  "losses": "risks of m training source domains R_1...R_m",
  "mean": "average risk R_avg across domains",
  "penalty": "variance risk R_var across domains",
  "nll": "cross-entropy loss for each domain"
}

**CONTEXT CODE:** 
# a FcNet 
class FcNet(nn.Module):
    def __init__(self,in_size:int = 4096, mid_size:int = 512):
        """
        A full connected network
        
        Args:
            args (object) : args object generated by opt_util.py
            mid_size (int) : The bottleneck layer size of the network
        """
        super(FcNet, self).__init__()
        self.desen_layer = nn.Sequential(
            nn.Linear(in_size,in_size),
            nn.Dropout(),
            nn.Linear(in_size,mid_size)
        )
        self.in_features=mid_size
    def forward(self,x):
        x = self.desen_layer(x)
        return x
    
def get_fea(args):
    return FcNet(in_size=len(args.HVG_list))

class feat_classifier(nn.Module):
    def __init__(self, class_num, bottleneck_dim=512):
        """
        A classification layer

        Args:
            class_num (int) : Number of classes.
            bottleneck_dim (int) : The input size of the classification layer
        """
        super(feat_classifier, self).__init__()
        self.fc = nn.Linear(bottleneck_dim, class_num)
    def forward(self, x):
        x = self.fc(x)
        return x

class VREx(nn.Module):
    def __init__(self, args):
        """
        The network contains a feature extractor and a classifier
        Args:
            args (object) : args object generated by opt_util.py
        """
        # MASKED: Initializes the VREx neural network architecture with feature extractor and classifier modules for Cancer-Finder implementation
        raise NotImplementedError("This function is masked for testing purposes.")

    def update(self, minibatches, opt, sch):
        """
        Network forward propagation, back propagation, parameter updating

        Args:
            minibatches (list) : mini batch of training data
            opt (object) : optimizer of the network
            sch (object) : learning rate scheduler

        """
        # MASKED: Implements the VREx (Variance Risk Extrapolation) domain generalization algorithm with risk extrapolation loss function for Cancer-Finder training implementation
        raise NotImplementedError("This function is masked for testing purposes.")


    def predict(self, x):
        return self.network(x)


Complete the function implementation replacing "raise NotImplementedError".