Outsourced training needs a referee that re-runs one step and accepts the worker's result if it lies within a tolerance tau of the referee's own. Floating-point differences make some tolerance unavoidable, and any tolerance is room to skip work. The open question in the verifiable-training-mechanisms programme is what that room costs the requester.
An earlier step measured it on one small network. This one repeats it on a 64-128-64-10 MLP (digits) and a 30-32-2 MLP (breast cancer), 20 seeds each, plain SGD, and a worker who skips every step the referee would accept as a skip.
- Digits, deeper net. At tau = 0.01 (about the worst float16 drift) the worker skips 37% of steps and gives up 0.0025 of held-out loss, 0.6 seed standard deviations. At 0.03 it gives up 1.7 standard deviations.
- Breast cancer. Skipping lowers held-out loss up to tau = 0.03 (91% of steps skipped), and hurts (5.7 standard deviations) only at tau = 0.1. Skipping late low-norm steps acts like early stopping on an easy task, so the requester sees no quality loss and the cheater saves the work.
I predicted the breast-cancer threshold would be lower; it was not, and the sign was opposite. The tolerance that is safe depends on the task's update-norm distribution, so it should be stated relative to a norm quantile. Caveats: small tabular and image tasks, one fixed train/test split, no transformer yet. Code, tests and the log: research/verifiable-training-mechanisms. AI-assisted.