I mean: the Huber/White/sandwich estimator of standard errors. It seems to me that, in the case of continuous outcomes, robust estimators of standard errors are rather simple, given that variance of residuals for each observation is calculated as the squared (estimated) residuals from the regression. But I can't figure out how this apply to a binary outcome: after estimating a probability for each observation, how can I estimate the variance of residuals for that observation, given it will necessarily have an outcome of 0 or 1, corresponding, in the logit scale, to either $+\infty$ or $-\infty$? In particular, I'm thinking at the case where I have continuous predictors (so I just have one occurence for each value of my set of regressors).
PS: I read some criticisms about the use of robust standard errors for logistic regression, because, if the estimates of variances are biased, then also the parameter estimates themselves are (given average and variance are linked, in the binomial case). However, I'm not wondering about whether robust standard errors make sense here, but simply how they are calculated.