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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.

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