# Loss function for a risk neutral binary classification

For binary classification task, with samples labeled $$y=0$$ and $$y=1$$, a neural network has one output node with sigmoid activation function, producing predictions $$\hat{y}\in(0;1)$$. Is the following loss function usable or is it flawed (mathematically or numerically)? $$L(y, \hat{y}) = \begin{cases} 1-\hat{y}, & \text{if y=1} \\ \hat{y}, & \text{if y=0} \end{cases}$$

The final objective is to predict $$1$$ (take an action) only when confident enough - "cherry-picking". Predicting $$0$$ incurs no cost but also has a zero utility. I think that translating it straight into a loss function woudn't help training a neural network, just like 0-1 loss.

I started with binary cross entropy and also considered using focal loss. However, the lowest achievable error rate is high (just below $$\frac{1}{2}$$), it is just not feasible to make a confident prediction. BCE seems to overly penalize being even appropriately somewhat confident. So I thought - why not trying a loss function which is even closer to the quantity I truly want to optimize.