# Deep Network with constraint or auxiliary features

The target of my current neural network is to predict a label. The dataset contains some features, there is a label $$y_i$$ in transaction $$i$$, indicating its classification. There is one feature $$f^{i}_j$$ can be used while training and is not available in deployment (this is very common in real-world applications). I consider this feature as a constraint value because the real label $$y_i$$ must be subject to some constraint function, for example, $$y_i <= \mathbf{C}(f_j)$$ where $$\mathbf{C}(\cdot)$$ is a constraint function.

My question is if I consider it as a constrained optimization problem? How can I get started? Could you please provide some helpful papers? Or if I consider the feature as an auxiliary feature, how can I leverage it?

Another perspective is to consider the $$f^{i}_j$$ as a prior, and the target is to maximize the posterior of the label prediction $$\hat{y}_i$$.

Thank you very much.