Anomaly Detection/Novelty detection
I believe you can use a classification algorithm where you manually overrepresent the "anomalies" class. By how much, depends on the cost induced by the anomalies.

Just to illustrate what I mean:

Anomalies cover a continuum between two extremes: 1) those which can be safely ignored, because they induce no costs, and 2) those whose costs is unbearable. As an example close to the extreme (1), imagine a flower shop getting an order for Iris virginica. The customer will hardly object if, among 6 million flowers shipped, 50 actually belong to the species Iris versicolor.

On the other hand, if you are shipping airplane parts, and a faulty part would cause a disaster with 300+ dead, then you are closer to the extreme (2). You will likely want to avoid shipping even one faulty part.

In practice, it will have to be a trade-off: You can be on the safe side if you ship no parts at all, but then you go bankrupt. On the other hand, if you do no quality assurance at all, you ship enough faulty parts to cause a disaster in a foreseeable future, have to pay damages, and go bankrupt again.