# One sided label smoothing in GANs

How does one-sided label smoothing make the discriminator more robust by reducing the confidence in correct class?

If we replace positive classification targets with $$\alpha$$ and negative targets with $$\beta$$ the optimal discriminator $$D$$ becomes:
$$D(x)=\frac{\alpha p_{data}(x) + \beta p_{model}(x)}{p_{data}(x) + p_{model}(x)}$$.
Where the $$p_{data}(x)$$ corresponds to the original distribution; at least we think. And the $$p_{model}(x)$$ corresponds to the distribution that we want to target and $$x$$ are the input features to the discriminator $$D$$.
Now, when $$\beta \neq 0$$, erroneous samples from $$p_{model}$$ have no incentive to move nearer to the data i.e. it reinforces current generator behaviour. Hence, we smooth only the positive labels to $$\alpha$$, leaving negative labels set to 0. This behaviour prevents discriminator from giving very large gradient signal to the generator.