Penalising wrong answers only works if someone else is grading

Our earlier paper on catastrophic forgetting found that once you freeze one expert per task, almost all of the difficulty is choosing which expert to trust. Each Split-MNIST expert is about 98.7% accurate on its own pair of digits, but when the system picks whichever expert sounds most confident, a wrong expert wins on roughly 37–39% of inputs. The obvious analogy is a multiple-choice test with no penalty for wrong answers: if guessing costs nothing, everyone guesses. So we asked whether penalising wrong answers, or anything else, would stop the confident wrong votes, and ran ten pre-registered experiments over five seeds.

The first was a check of calibration, whether an expert that says 70% is right 70% of the time. On its own task it was close to perfect, with an expected calibration error of 0.011. On every other task's inputs the error was 0.697: of the inputs an expert scored at 95% confidence or higher, only 34% belonged to its task. Rescaling every expert by one temperature changed nothing, as it must, because it cannot change which expert is most confident. A separate calibration map for each expert helped by 3 points.

The penalty idea is formalised by Kalai and colleagues: announce a confidence target t, score a wrong answer at −t/(1−t) and an abstention at zero, and answering only above t becomes the best strategy. We gave each expert an abstain output and trained it with a wrong-answer penalty on its own data. It failed, and at stricter targets it made things worse: wrong-task selections rose 54% at t = 0.75, because the abstain output fired on 18% of the expert's own inputs and 20% of everyone else's. An expert that has never seen another task cannot learn to abstain on it. Cross-entropy already penalises confident mistakes, but only on the data it is trained on.

Give the expert that data and the idea works. Training a "not mine" output on 100 stored images per task lifted accuracy from 0.633 to 0.871, close to what Henning and colleagues reported in the same setting. With no stored images at all, training against rotated and flipped copies of the expert's own digits, the trick behind Kim and colleagues' CSI-based method, recovered about half of that gain.

The largest gain came from not asking the experts at all. Choosing the task with a statistical model of shared, drift-free features, RanPAC's random projection and ridge classifier, reached 0.926 ± 0.004, against 0.786 for selecting by confidence over the same features, 0.857 for nearest class mean and 0.859 for the GDumb baseline. It also degraded least as tasks were added. That made the pre-registered kill on the "not mine" experiment fire: even with 100 stored images per task, its error rate stayed above that of the selector that stores none.

The penalty belongs at the selector, applied to a score the experts cannot inflate. With a reject threshold on the margin of a density model, the Kalai score at a 90% target was +0.545 while still answering 71% of inputs; the same threshold on expert confidence scored +0.114. Two other results were simpler. Deciding the task over 16 inputs known to come from the same task brought wrong selections to zero for every selector. And a gate that only compares each input with each task's own training data, by how well that task's principal components reconstruct it, reached 0.915 on raw pixels, though that arm was secondary in the pre-registration.

These are small numpy MLPs on Split-MNIST, five seeds, with no correction across roughly forty arms. The shared features are fit on the whole unlabelled stream, which a strict continual learner would not have, and RanPAC's one hyperparameter was tuned on a labelled calibration split. Two experiments fired their kills and one prediction about how errors scale failed: wrong experts tend to steal the same inputs rather than a few look-alike task pairs. The paper and the findings ledger record all ten results.