Custom Loss Function - Inducing sparsity

From the comments, I realized that my question wasn’t clear enough, so I’ll start with a short background.

I am trying to construct an attention model that performs classification based on just a small region of the original image (as small as possible).

For this I have one branch of my model output a binary mask of 0’s and 1’s which is than multiplied element-wise with the original image before the masked image is passed on to the classification branch. The output of the attention branch is a map of HxWx1 taken after a hard sigmoid activation. The architecture is demonstrated in the following figure:

I do not have the GT of the attention mask. This part of the model is trained in an unsupervised manner, but I’m trying to avoid using reinforcement learning.

How do I define a loss function that at the same time:

1. Regresses the outputs to be either 0 or 1 (or very close to these values).
2. Attempts to reduce the number of 1’s and increase the number of 0’s.
3. Is still a differentiable function.

Attempts:

1. I initially tried using MSE loss between the mask and a zero mask, but that didn’t work (the output is a mask of small non-zero values of similar magnitude). It didn’t have any incentive to use high values or reduce the number of non-zero elements.

2. I than moved on to using the following loss function on the mask output:
   

This loss is summed with the standard cross-entropy loss of the classification output for the total loss of the model:

I_xy is the value of the mask at pixel (x, y). Parameter α∈[0,1] is the sparsity regularizer, the higher its value, the higher the incentive of the model to reduce the number of active pixels. Parameter β is the mask-regularization parameter and it determines the contribution of the mask loss to the total weighted sum (the weight of the classification branch is kept on 1.0) and λ is small coefficient added for numerical stability.

However, the training with this loss function is very unstable (mask tends to zero out) and I don’t feel like I am going the right way.