 Solution

For conditionally independent binary labels, the likelihood isMaximizing it is equivalent to minimizing its negative log-likelihood, the mean binary cross-entropy:It is the loss implied by a Bernoulli distribution for each target, penalizes confident errors strongly and is convex in the linear-logistic parameters. For numerical stability, each contribution can be evaluated as $\log (1+e^{u_i})-y_i{u}_i$ using a stable softplus implementation.

The 10:90 class imbalance does not invalidate this likelihood. Class weighting or resampling may be useful if errors have unequal costs, but changes the training objective and potentially the interpretation of fitted probabilities. Optional ridge regression adds $\lambda \Vert \mathbf{w}{\Vert }^2/2$, usually excluding the intercept, to control overly large weights or complete separation.